[Yuan Wang] the Goldbach Conjecture(Bookos.org)

Series in Pure Mathematics - Volume 4

Goldbach Conjecture

Editio

Yuan Wang

World Scientific

The

Goldbach Conjecture

Second Edition

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Series in Pure Mathematics - Volume 4

The

Goldbach Conjecture

Second Edition

Yuan Wang Academia Sinica, China

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THE GOLDBACH CONJECTURE — Second Edition

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PREFACE TO THE SECOND EDITION

Since the publication of Goldbach Conjecture in 1984, much progress has been made in the study of Goldbach Conjecture. Many friends persuaded me to publish a revised edition containing these new materials. Recently I received a letter by Dr. Stanley Liu from World Scientific Publishing Company asking me to do the same thing. I think it is a good occasion for me to give several supplements of the book.

In the new edition, we add a proof of (2,3) by A. Selberg and a proof of Three Primes Theorem for all odd numbers > 9 under the assumption of (GRH) by J. M. Deshouillers, G. Effinger, H. Te Riele and D. Zinoviev. Besides there are a few changes in the original Introduction and supplements on the References.

Finally, I should like to express my thanks to Professors Pan Cheng Biao, Liu Ming Chit, Wang Tian Ze, Wang Yong Hui and Liu Jian Ya for their valuable help and to Dr. Stanley Liu and Dr. Jitan Lu of World Scientific Publishing Company for their generous cooperation and assistance.

Wang Yuan

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PREFACE

The study of Goldbach conjecture has had wery great achievements

since 1920. In particular, I. M. Vinogradov proved in 1937 the

three primes theorem and Chen Jing Run established the (1, 2) in

1966. Furthermore, we must point out that the investigation on

Goldbach conjecture has given a tremendous impetus on the creation

and development of many powerful number-theoretic methods which are

very useful not only in number theory itself but also in many other

fields of mathematics.

The three primes theorem and the (1, 2) have been collected in

many books (see Refs. I). A monograph which often contains the

latest results with possibly simplified proofs so that the reader

can easily understand, is difficult to contain the main steps on the

development of original ideas, however. The aim of the present

collection is to select as far as we can the papers with origination

and progress in techniques so that the reader can understand the

major steps of the whole progress in the study of Goldbach conjec

ture. We hope it will be of benefit for further studies of this

problem.

In order that the volume will not be too thick, some parts in a

few papers are deleted, where the Editor's notes are given for expo

sition. All papers in Chinese, French, German and Russian have been

translated into English.

I should like to express my gratitude to Professor Pan Cheng Biao

and his students, and Professor Yu Kun Rui for their valuable assis

tance, and I must also thank Dr. K. K. Phua for his help in the

publication of this collection.

October 1983 Wang Yuan


CONTENTS

Preface to the Second Edition v

Preface vii

Introduction 1

I. Representation of An Odd Number as the Sum of Three Primes 1. Some problems of "partitio numerorum";

III: on the expression of a number as a sum of primes

G.H. Hardy &z J.E. Littlewood 21

2. Representation of an odd number as a sum of three primes

I.M. Vinogradov 61

3. A new proof of the Goldbach-Vinogradov theorem Ju. V. Linnik 65

4. A new proof on the three primes theorem C.B. Pan 72

5. An elementary method in prime number theory R. C. Vaughan 81

6. A complete Vinogradov 3-primes theorem under the Riemann hypothesis

J.M. Deshouillers, G. Effinger, H. Te Riele & D. Zinoviev 91

II. Representation of An Even Number as the Sum of Two Almost Primes (elementary approach)

7. The sieve of Eratosthenes and the theorem of Goldbach

V. Brun 99

8. New improvements in the method of the sieve of Eratosthenes

A. A. Buchstab 137

9. On prime divisors of polynomials P. Kuhn 154

10. On an elementary method in the theory of primes A. Selberg 157

11. On the representation of large even number as a sum of two almost primes

Y. Wang 161

12. Lectures on sieves A. Selberg 166

III. Representation of an Even Number as the Sum of a Prime and an Almost Prime

13. On the representation of an even number as the sum of a prime and an almost prime

A. Renyi 185

14. On the representation of large integer as a sum of a prime and an almost prime

Y. Wang 192

15. On representation of even number as the sum of a prime and an almost prime

CD. Pan 214

16. The "density" of the zeros of Dirichlet L-series and the problem of the sum of primes and "near primes"

M.B. Barban Til

17. New results in the investigation of the Goldbach-Euler problem and the problem of prime pairs

A.A. Buchstab 238

18. The density hypothesis for Dirichlet L-series A.I. Vinogradov 245

19. On the large sieve E. Bombieri 249

20. On the representation of a large even integer as the sum of a prime and the product of at most two primes

J.R. Chen 275

21. A new mean value theorem and its applications CD. Pan 295

References I 309

References II 311

INTRODUCTION

In a letter to Euler in 1742, Goldbach proposed two conjectures on the representations of integers as the sum of primes. These conjectures with some modifications may be stated as follows.

(A) Every even integer > 6 is the sum of two odd primes. (B) Every odd integer > 9 can be represented as the sum of three

odd primes.

Clearly, (B) can be derived from (A). In his letter to Goldbach, Euler expressed his belief in these state

ments, though he could not prove it (see Dickson [1]). It is shown that these two conjectures are correct by a lot of

accumulated numerical calculations since Goldbach wrote his letter up-to-date, for example, Shen Mok Kong [1] has checked the conjecture (A) up to 33 x 106, and the further calculation up to 108 was made by Light, Forres, Hammond and Roe [1]. The latest result in this direction was obtained in 1998 by Saouter [1] who established that (B) is true for each odd integer < 1020.

In his famous speech at the 2nd International Congress of Mathematics held in Paris in 1900, Hilbert [1] proposed 23 problems for the mathematicians in the 20th century, and the conjecture (A) is a part of his 8th problem. In 1912, the conjecture (A) is regarded as one of the four famous unsolved problems in the theory of prime numbers proposed by Landau [2] in his speech at the 5th International Congress held in Cambridge. Furthermore, in his speech at the mathematical society of Copenhagen in 1921, Hardy [1,2] pronounced that the conjecture (A) is "probably as difficult as any of the unsolved problems in mathematics" and therefore Goldbach problem is not only one of the most famous and difficult problems in number theory, but also in the whole of mathematics.

There is no method to attack this problem and the research is confined only on checking the conjecture (A) by some numerical

calculations or proposing some further conjectures on (A) since Goldbach wrote his letter up to 1920 (see Dickson [1], Hardy [1,2]).

The first great achievements on the study of Goldbach problem were obtained in the 1920s. Using their "circle method," British mathematicians Hardy and Littlewood [2] proved in 1923 that every sufficiently large odd integer is the sum of three odd primes and almost all even integers are sums of two primes if the grand Riemann hypothesis is assumed to be true. Norwegian mathematician Brun [2,3] established in 1919 by his "sieve method" that every large even number is the sum of two numbers each having at most nine prime factors. And in 1930, by using Brun's method with his own new idea, the "density" of integer sequence, Russian mathematician Schnirelman [1] first obtained a theorem in additive prime number theory, namely every integer > 2 is the sum of at most c primes, hereafter we use c, ci, C2,... to denote absolute constants, but not the same constants at different occurrences. The study of Goldbach problem has a tremendous and deep development for the past eighty years. In particular, in 1937, using the circle method and his ingenious method on the estimation of exponential sum with prime variable, Russian mathematician I. M. Vinogradov [3] was able to remove the dependence on the grand Riemann hypothesis, thereby giving unconditional proofs of the above two conclusions of Hardy and Littlewood. And after a series of important improvements on Brun's method and his result, Chinese mathematician Chen Jing Run [2,3] established in 1966 that every large even integer is the sum of a prime and a product of at most two primes.

We must mention that the breakthrough on the study of Goldbach conjecture is clearly inseparable from the great achievements on analytic number theory in the 19th century, in particular, the theory of Cebysev, Dirichlet, Riemann, Hadamard, de la Vallee Poussin and von Mangoldt on the distribution of prime numbers which is the prerequisite of the present research.

Now we sketch the main ideas and progresses on the study of Goldbach conjecture as follow.

1. Circle method. The circle method has its genesis in a paper of Hardy and Ramanujan [1] in 1918 concerned with the partition

function and the problem of representing numbers as the sums of squares. More generally, in a series of papers beginning in 1920 entitled "Some problems of 'partitio numerorum'," Hardy and Littlewood create and develop systematically a new analytic method, the circle method in additive number theory, where Goldbach problem is devoted in (III) and (V) of the series.

Let OO -I

*—i. ns

n = l

When a < 1, ((s) may be defined by analytic continuation, ((s) is called the Riemann ^-function. Riemann conjectured that all zeros C = j3 + it of (,(s) on the half-plane a > 0 lie on the line a = \. This is an unsolved problem and it is denoted by (RH). The weaker conjecture that every p on a > 0 has its real part < 9, where 9 is a constant satisfying | < 9 < 1, is called the quasi Riemann hypothesis which is denoted by (QRH). More generally, we may study the Dirichlet L-function

£(*>X) = X! • > s = a + it, < J > 1 , 7 1 = 1

where x{n) is a character modq. It is regular in s-plane if x ¥" Xo> where xo denotes the principal character. Otherwise it has an only pole at 5 = 1 and L(s, xo) — n p | g ( l — ^)C(S)> where p denotes a prime number. Similar to (RH) and (QRH), we may define (GRH) and (QGRH), namely all zeros of all the L(s, x) on a > 0 lie on a — , and every zero p of any L(s, x) on a > 0 satisfies j3 < 9, where 9 is a constant as above. The two results of Hardy and Littlewood are based on the assumption of (QGRH) with 9 satisfying ^ < 9 < | .

Hereafter we use p, p', p\, p2,... to denote prime numbers. Let n be an integer > 1. Let

/(x) = X>SPK> (!) p>2

where |x| = e~xln. Then OO

/W3 = ^ r 3 ( n K , n = l

where r3(n) = Yl loS Pi loS P2 log Vz (2)

Pi+P2+P3=n

denotes a weight sum of the representations of n as the sums of three primes. We may define similarly r2(n), and thus the conjectures (A), (B) may be stated as follows:

r2(n) > 0 (2|n, n > 4) and r3(n) > 0 (2|n, n > 7).

By Cauchy's integral formula, we have

rs(n) = ^-.^ftx)3x-n-1dx, (3)

where T is a circle with center O and radius e - 1 / n . Since f(x) is approximated closely by / (e - 1 / / "e( - ) ) , where e(y) = e2my and a;(£r) is a nearby point of e - 1 / "e ( - ) ) , the T is divided into the sum of small arcs £/i9, where the amplitude of x on £hq lies between

( ; 7T)27r a i " i ( - + - ; TTT ) 2TT (modi) \q q{q + q')J \q q{q + q")J V '

in which K, ^, \T are three consecutive terms of the Farey series of order N — [y/n\. Hence

^ ) = E E i l [ Hxfx-^dx, (4) 9=1 h(q) Z m J^*

where h runs over a reduced residue system modq. When x E £hq, set

x = e ( - ) e~Y , Y = r} + iO. A,

Then under the assumption of (QGRH) with \ < 6 < | , Hardy and Littlewood established that

/(x) = 0 + $, (5)

where

<t> = 'mY and * = <V+*( log")C).

in which fj,(q) and cp(q) denote respectively the Mobius function and the Euler function. Substituting (5) into (4), we have

and therefore the proposition (B) holds for 2\n and n is sufficiently large. More precisely, it is easily derived from (6) the asymptotic formula of Rz(ri), the number of representations of the odd number n as the sums of three primes, namely R^{n) is asymptotic to r3(n)(log n)~3. But the circle method failed in treating ^ ( n ) , even the (GRH) is assumed to be true. The main difficulty is not in the principal term but in its error term. Hence if the term $ in (5) is neglected, that is, <f> is assumed to be used instead of / , then we have

p>2

It follows from (7) that the number of representations R2(n) of the even number n as the sum of two primes is asymptotic to r2(n)(log n)~2 . This is the famous conjecture of Hardy and Littlewood [2] concerning the Goldbach conjecture (A).

Under the (GRH), Hardy and Littlewood proved that

£ m = 2

2|m

-w-nO-^nHr' = 0(nb'2+e),

V p>2 / (8)

hereafter we use e to denote any preassigned positive number and the corresponding constant implied by the symbol o depends on e only. Let E{n) denote the number of even numbers less than n such that (A) is false. Then it yields from (8) immediately

E(n) = 0{n^+£), (9)

and consequently, we have the conclusion that almost all even integers are the sums of two primes.

Later, Vinogradov introduced a number of notable refinements on the circle method, one of which is to replace f(x) by the finite sum

F(a)= Y, e M - (10) 2 < p < n

The trivial orthogonality relation

f1 , , f l , whenfc = 0, , , / e(ak)da =\ (11)

Jo [0. otherwise v ' gives

Rz(n) = Yl l = I F(afe(-an)da (12) Pl+P2+P3=n °

which is used instead of (3). Vinogradov's modification has its genesis in a paper on Waring's

problem in 1928 (Cf. Vinogradov [1]). Let r = n_1(log n)Cl and Q = (log nf2. When q < Q, let

Mhq = h h

r, - + r 1 Q

, (h, q) = 1

which is called a "major arc". When n is sufficiently large, the major arcs are disjoint. The union of all M.hq is denoted by

M= u U' > Kq<Q h(q)

and its complement with respect to [0,1] is called the "minor arcs" which is denoted by m. Hence we have

R^n) = / F(a)3e(-an)da+ / F(a)3e(-an)da = I + J say ,

(13)

and thus the proof of (B) for large n is reduced to prove that I is the principal term of Rz{ri) and J gives only of lower order.

Remark. The distinction between major and minor arcs was first proposed by Hardy and Littlewood in their work on Waring problem.

The difficulty for the estimation of I can be removed by the following Siegel-Walfisz theorem.

Let q < Q and (h, q) = 1. Then 1 fX (If ,

*<*• «. *> = E 1 = W} J, j ^ + CC-VST!), (14) p=/i(log g)

where the constant implicit in O depends on C2 only (Cf. Siegel [1], Walfisz [1].)

From (14), it gives

Hi) ^ 2 ^ g ^ ?

Substituting (15) into the expression of I, we have

I T T A 1 \ T T / - 1 2

'-inti-snpjne+srô^. 2t"- <16)

Hence the difficulty is concentrated to the estimation of F(a) when a G m. In 1937, using his creative and ingenious method on the estimation of exponential sum with prime variable, Vinogradov gave F(a) a non-trivial estimation, namely

F(a) < n(log n)~c, a Em, (17)

where c is a constant > 3. Notice that for given c, we may take c\, C2 to be the constants depending on c. It follows by (17) that

J < n(log n)~ c / \F{a)\2da<^n2{\ogn)'A. (18) Jo

Substituting (16) and (18) into (13), we have

^ ~ i n ( , - ^ i ? ) n ( » + 5 ^ ) o ^ . 2'-and thus we have proved that there exists a constant no such that every odd integer n(>no) is the sum of three odd primes. This theorem is called the Vinogrado-Goldbach theorem or the three primes theorem.

We should mention that the following two theorems had appeared before the proof of three primes theorem, (i) Every large odd number n can be represented as

n = Pi + f2 + P3P4 (19)

(Cf. Vinogradov [2], Estermann [2].) (ii) Every large number is the sum of two primes and a square (Cf. Estermann [3].)

If the Page theorem [1] is used instead of Siegel-Walfisz theorem on the evaluation of / , the no in three primes theorem is calculable, and it was given by Borozdkin [1] in 1956 that no = ee = JQ6,846,168.5... Because of the importance of Vinogradov's result, we call the value of n0 the Vinogradov bound, no was improved by Chen Jing Run and Wang Tian Ze [1] in 1989 to n0 = e6"'503 = io43-000-5

and finally Liu Ming Chit and Wang Tian Ze [1] established in 2001 that n0 = e3'100 = io1-346-3-. Another direction is to calculate no under the assumption of (GRH). Lucke [1] in 1926 obtained no = IO32 and finally Deshouillers, Effinger, Riele and Zinoviev [1] proved no = 9 under the assumption of (GRH). That is, they proved the whole conjecture (B) under (GRH).

Using Vinogradov's method, several mathematicians have pointed out independently that almost all even integers are the sums of two primes. More precisely, they proved that for any given c,

£ ( n ) < n ( l o g n ) - c , (20)

where the constant implicit in <C depends on c only. (See van der Corput [3], Estermann [4], Heilbronn [1], Hua [1], Tchudakov [1,2].)

In 1946, using the original method of Hardy and Littlewood such that T is divided into M and m, Russian mathematician Linnik [3,4,5] gave f(x) (x € m) a non-trivial estimation similar to (17), and therefore he gave a new proof for the three primes theorem. Linnik's estimation of f(x) is based on his important density theorem for L-series which is used instead of the unproved (QGRH) namely.

Let x(n) be a primitive character modq. Let N(/3, T) denote the number of zeros of L(s, x) in the rectangle

" < < r < l , |*| < T ,

where T > q50, /3 > 1 and v = /? - \. Then

N(P, T) < ^ ^ - ^ - ^ ( l o g T)w + qso (21)

(Cf. also Tchudakov [3].) Later, Vaughan [2] gave a simplified proof on the Linnik's estima

tion of f(x) in 1975, and a further simple proof is given by Pan Cheng Biao [1] independently in 1977. In their proofs, only some elementary knowledge of L-function are used. Concerning the estimation of exponential sum F(a), Vaughan [5,6] gave also some modifications, and his idea is the use of identity

— = -j{l-LG)-L'G= (-— - F)(1-LG)-L'G+F-LFG

which is also pointed out by Pan Cheng Dong (see Pan Cheng Dong, Ding Xia Qi and Wang Yuan [1].)

By the application of further results on the density theorems of L-function, Vaughan [1] proved in 1972 that

£(n) « n e - V ^ . (22)

Later, (22) was improved by Montgomery and Vaughan [1] in 1975. They proved that there exists a constant 6 such that

E{n) < n1'5 . (23)

Chen Jing Run and Pan Cheng Dong [1] pointed out that 5 > 0.01, and it was improved further by Chen Jing Run [5] to 5 > 0.04.

Besides, Linnik [6, 7,8, 9] was the first, who proved the following two important theorems by the use of circle method, namely (i) for any given positive integer g > 1, there exists &o > 0 such that for k > ko, every large integer = kg (mod 2) can be represented by

n = Pl+p2 + 9xl + ---+9X\ (24)

where xi,..., x^ are positive integers, and (ii) under the assumption of (RH), for any given integer n > 1, there exist pi, p2 such that

| n - p i - p 2 | « ( l o g n ) 3 + £ . (25)

Concerning these two problems, A. I. Vinogradov [1], Gallagher [4], Katai [1], Montgomery and Vaughan [1], Wang Yuan [7], Prachar [3], Pan Cheng Dong [5], Lu Min Gao [1], Wang Yuan and Shan Zun [1], Liu Jian Ya, Liu Ming Chit and Wang Tian Ze [1, 2], Wang Tian Ze [1], Li Hong Ze [1], D. R. Heath-Brown and J. C. Puchta [1] have made some valuable contributions, for example, Katai proved that the right hand side of (25) can be replaced by (logra)2 and Heath-Brown and Puchta establish that k may be taken by 13 in (24) if g = 2, and it may be reduced further to 7 under the (GRH).

2. Sieve method. The historical origin of sieve method may be traced back to the "sieve of Eratosthenes" in 250 B.C. Eratosthenes noted that the prime numbers between ^/n and n can be isolated by removing from the sequence 2, 3 , . . . , n every number which is a multiple of a prime not exceeding ^/n. Let ir(x)(— ir(x, 1, 1)) denote the number of primes < x and n = Ylp<îP- Then

l + 7r(n)-7r(v^) = £ £ »{d) a<n d\(a,Tl)

d|n (26)

If we use ^ + #(—1 < 6 < 0) instead of [^], then it will cause an error term

0 ( 2 * ^ ) (27)

in (26), and so the sieve of Eratosthenes with its large error term compared with n is almost useless.

It was a great achievement when Brun [3] in 1919 devised his new sieve method and applied it successfully to several difficult and important problems in number theory, in particular, the Goldbach problem. In 1947, Selberg [2] gave another sieve method which leads to more precise result than Brun's method in every known case, when it can be applied. Moreover, Selberg's upper bound method is surprisingly simple, and it has a certain air of finality. Indeed these methods represent the indispensable tools in number theory.

The essence of the methods of Brun and Selberg is to use some inequalities instead of

A / N v-^ / « f 1> when n = 1, , v

A(n) = !>(«) = {„ otherwjs • (28) d\n \ ' '

in order to decrease the error term in Eratosthenes sieve. Brun defined two sets of integers D\ and D2 such that

£ M(d) < A(n) < £ /*(<*) • (29)

However, the construction of D\ and D2 is complicated and heavily-combinatorial in character. Selberg noted that

(30)

holds for any set of real numbers \'ds with Ai = 1. Choose suitable X'ds. Then we obtain the Selberg's upper bound method. Selberg [2,3,4] published only his upper bound method, and indicated its place in the construction of lower bound sieves, without ever publishing any details. Selberg's idea was developed and accomplished by Wang Yuan [1, 2], A. I. Vinogradov [2,3], Levin [1, 2], and Jurkat and Richert [1].

Let A = {au} be a finite set of integers. Let P denote a finite set of primes. Further let F(A, P) be the number of elements in A which is unsifted by the sequence P. Take av = v(n — v) (1 < v < n) and P the set of all primes < n1/^"1"1), where 2|n and t is a natural number. Denote F(A, P) — F(n, n1/^1"1)). Suppose that we can obtain a positive lower estimation for F(n, n1/^"1"1)) when n is large. Then it follows that every large even integer n is the sum of two numbers each being a product of at most t prime factors, and we denote this proposition by (£, £). Similarly, we may define (£, m) for

Brun was the first who proved the (9,9). Brun's method and his result were improved by several mathematicians, namely (7, 7)

(Rademacher [1], 1924), (6,6) (Estermann [1], 1932), and (5,7), (4,9), (3,15) and (2,366) (Ricci [1,2], 1937).

The power of the methods of Brun and Selberg will be vastly improved if some combinatorial relations are used. These combinatorial ideas are of two kinds. One is the use of combinatorial identities for iteration originated in a paper of Buchstab [1] in 1937, and the other is the introduction of weighted sieves in 1941, of which the pioneering work is due to Kuhn [1].

Let F(A, q, q') denote the number of elements in A satisfying av = 0 (modq) and av ^ 0 (modp) (p < q'). Then

F(A,Ps) = F(A,Pt)- J2 ¥ . ? . ? ) • (31) Pt<p<ts

This is called the Buchstab's identity. From a lower estimation of F(A, pt) and those upper estimations of F(A, p, p), it yields a lower estimation of F(A, ps). Similarly, we may obtain an upper estimation of F(A, ps). To iterate by (31) successively, we may obtain better upper and lower estimations for F(A, P). By the use of Selberg's method, the explicit expressions of upper and lower estimations for certain F(A, P) were obtained by Jurkat and Richert [1] in 1965. Let F(A, b, q, q') be the number of elements in A such that au ^ 0 (modp) (p < q) and au satisfies at most b congruences of those av = 0 (modp') (q < p' < q'). Then

F(A, b, q, q') > F(A, q) - ^ £ F(A, p, q). (32) q<p<q'

Choose suitable q, q' and b. Then a positive lower estimation of F(A, b, q, q') often leads to a better result on Goldbach problem. Buchstab proved (5,5) [2] in 1938 and (4,4) [3] in 1940, where, (4,4) was also announced by Tartakovskii [1,2], and Kuhn [3] established (a, b) (a + b < 6) in 1954.

By the combination of the methods of Brun, Selberg, Buchstab and Kuhn, Wang Yuan was able to prove (3,4) [1] in 1956, and (3, 3), (a, b) (a + b < 5) [3] and (2,3) [4,5] in 1957, where (3,3) was also obtained by A. I. Vinogradov [2,3] in 1956 independently. (2,3) was announced by Selberg [3] and his proof was appeared in [5,6].

Another proofs of (2,3) along the line of Wang Yuan's argument were given by Levin [2,3] and Barban [4, 5] in 1963, and in their proofs, the numerical calculation is simple but the use of deep analytic method is required.

If we take A — {n — p, p < n) and P the set of all primes < n1/^"1"1), then a positive lower estimation for F(A, P) leads to (1, t). This set A was first introduced in 1932 by Estermann [1] who proved (1,6) by Brun's method under the assumption of (GRH). Wang Yuan [2] and A. I. Vinogradov [2,3] improved the (1,6) to (1,4) in 1956 under the same hypothesis, and Wang Yuan [3,6] gave further improvement (1,3) in 1957.

In order to remove the unproved hypothesis in the above results, it needs new idea and method. In the sieve so far considered, the sequence A is sifted, for each p e P, by the residue class 0 modp. But in some applications, it should sift, for each p E P, the sequence A by the residue classes

hp,i,..., /iPifc(p)(modp).

The methods of Brun and Selberg can be applied effectively to this case unless k(p) is, on the average, exceedingly small compared with p. Otherwise they are ineffective. In 1941, Linnik [1] devised an ingenious method, the so called large sieve. It yields a non-trivial upper estimation for the number of unsifted elements of A not exceeding n when k(p) is, on the average, comparatively large. Hungarian mathematician Renyi [1,2] improved Linnik's methods in various respects, and he proved successfully (1, c) in 1948. Further important improvements on the large sieve were obtained by Roth [1] and Bombieri [1] in 1965. In Renyi's paper, a mean value theorem for TV(X, k, h) is proved by means of large sieve that may be used instead of (QGRH) in the proof of (1, c), namely

7T(X, k,h)-^r =0 (n X . ) . (33)

Notice that ir(x, k, h) should be replaced by a weight sum in his original paper. If (33) holds with 5 = \ — s, it may be used

max (k,k)=l

instead of (GRH) in the proofs of the results due to Estermann, A. I. Vinogradov and Wang Yuan (Cf. Wang Yuan [6].)

In 1961, Barban [1] proved (33) with 5 = \ - e. In 1962, Pan Cheng Dong [2] established independently (33) with 8 = ~ — e, and derived (1, 5). In 1962, Wang Yuan [6] pointed out that (1,4) can be derived from 5 = \—£. Pan Cheng Dong [3] in 1962 and Barban [3] in 1963 proved independently that (33) holds for 5 = | — e, and derived (1,4) without the use of heavy calculations. In 1965, Buchstab [7, 8] established (1,3) by the use of 5 = | — £ with heavy calculations. Bombieri [1] and A. I. Vinogradov [4] proved independently (33) with 5 = \ — £, and so it follows easily (1,3). (33) with 6 = ^ — £ is called the Bombieri-Vinogradov's mean value theorem. More precisely, Bombieri established the following important formula

y ^ max 1 ^ {h,k)=l

fc<l2/(log i ) c 2

( 1 U\ l i X

*(x' k'h)-Ak) = ° l ( S ^ F ' (34)

where C\ is any given constant and C2 is a constant depending on c\. Although Bombieri's formula is only slightly stronger than (33) with 6 = \ — £, it has many important applications in number theory. For example, (34) can be used instead of (GRH) in the Hooley's proof of the following famous theorem. Let N(n) be the number of representations of n = p + u2 + v2. Then

l o g n ^ p ( p - i y p s l { I d 4 ) p2 - p + i

p\n

• n ^ r v P5) p = 3 ( m o d 4 ) ^ y

p\n

where x(n) is the non-principal character mod 4. Remark. Using their circle method, Hardy and Littlewood [2]

conjectured (35), but they failed to prove it even under the assumption of (GRH). In 1957, it was Hooley [1] who first gave an elegant proof of (35) under the assumption of (GRH), and then Linnik [10,11] proved (35) unconditionally by his complicated dispersion method in

1960. Another advantage of Bombieri's paper is his proof of (35) which is creative and simple. The further simplified proof of (35) is given by Gallagher [1,2].

In 1966, Chen Jing Run [2,3] gave an important improvement on weighted sieve, and so he proved (1,2) which is called the Chen's theorem, that is, every large even integer is the sum of a prime and a product of at most 2 primes. Let

M = N-Q + 0{n9/1°), (36)

where

iV = F(n, n1/10) -]- ] T F(n, p, n1/10) rel/10<p<„l/3

and

p < n n — p = P1P2P3 (Pl,2) P3<n/piP2

in which 2|n, A = {n — p, p < n} and (^1,2) denotes the condition n1/10 < pi < n1/3 < p2 < (^r)5 . Then a positive lower estimation for M when n is large implies (1,2). In fact, M > 0 means that there exists a prime p such that n—p has at most 1 prime factor in the interval [n1/10, n1/3] and 1 prime factor > n1/3, ox n—p has only prime factors > n1/3 , and so the assertion follows. N in the right hand side of (36) is given by Kuhn's inequality (32) with suitable choice of parameters (see Wang Yuan [6]), and it may be estimated by the methods of Brun, Selberg and Buchstab combined with the Bombieri-Vinogradov's formula. Chen Jing Run's ingenious idea is the introduction of Q, and gave it a non-trivial estimation. All the later simplified proofs on Chen's theorem are concentrated on the simplification of the estimation of fi, in particular, Pan Cheng Dong, Ding Xia Qi and Wang Yuan [1] pointed out that the estimation of fi can be derived immediately by the following mean value theorem similar to (34).

Let 2 < y < x and 7r(y, a, 9, h) — Y^ °-v<y 1- Then ap=/i(mod 9)

V^ max max ^ V<X (h,q)=l

5<x2 / ( log x)c2 C3<0<C4

7r(y, a, q, h) liy-

4>{q)

= O 77— - r - (37) \ ( logx) c i ' v '

holds uniformly on C3, C4 satisfying (logy)2c2 < c% < C4 < y1 _ £ , where | / (a ) | < 1, c^ = ci + 7 and the constant implicit in O depends on e and c\.

Furthermore, Pan Cheng Dong and Ding Xia Qi [1, 2] established a mean value theorem which includes (34) and (37).

3. Density. Let A denote a set of distinct non-negative integers with its element denoted by o. Let A(n) = Xa<a<n 1. Further let a = inf„>i —^ which is called the Schnirelman density of A. Clearly 0 < a < 1, and a = 1 means that A contains all natural numbers. Similarly we may define B, b, B(n), f3 and C, c, C(n), 7. The sequence of all distinct numbers with the form of a + b(a G A, b 6 B) will be denoted by C = A + B. We define 2A = A + A and sA = A + (s — 1)A by induction. Schnirelman [1,2] established two simple but important theorems, namely (i) if 0 € A and 1 € B, then 7 > a + f3 - a/5, and (ii) if 0 E i , 1 G B and a + f3 > 1, then 7 = 1, that is C contains all natural numbers. It follows from (i) that if a > 0, then there exists an integer s0 such that the density of so A is > | , and consequently, 2SQA contains all natural numbers by (ii), that is, (iii) if 0 G A and a > 0, then every positive integer can be represented as the sum of 2so elements of A. Let A* be a set of non-negative integers in which the element is allowed to be repeated. Let A be the set of all distinct elements in A* and r(a) denote the number of repetitions of a in A*. Then by Schwarz inequality, we

have

£ r(a)\ < £ Haf £ 1 v l<a<n / l<a<n \<a<n

= A(n) £ r(af, Ka<n

and therefore

« > £ r(a) n £ r(a)2 . (iv) \ l < a < n J I l<a<n

The concept of Schnirelman's density is certainly simple but it is very useful. Let r(a) denote the number of representations of a = pi + p2. Then Brun's method yields

Take A* be the set contained 0, 1 and the numbers of the form a = pi + p2. Then it follows by (iv) that A has positive density. Hence we have by (iii) the famous Schnirelman-Goldbach theorem, namely there exists a constant c such that every integer greater than 1 is the sum of at most c primes.

Let s be the smallest integer such that every large integer is the sum of at most s primes. Then Schnirelman's original method implies that s < 800,000. Since some results on Schnirelman's density and Brun's method are further improved, in particular, Khintchine [1] proved in 1932 that 7 > min(l, 2a) if A = B, Mann [1] established in 1942 the famous a + j3 conjecture, that is, 7 > min(l, a + /3), and Selberg devised his new sieve method, the estimation of s is improved also, namely s < 2,208 (Romanov, 1935 [1]), s < 71 (Heilbronn, Landau and Scherk, 1936 [1]), s < 67 (Ricci, 1936 [1,2]) and so on. The best record s < 6 is due to Vaughan [3]. However the precision of this result is still inferior to s < 4 implied by the three primes theorem. By Schnirelman's method, we may estimate also

the smallest integer s such that every integer > 1 is the sum of at most s primes.

Although the three primes theorem and the (1,2) are inferior to (1,1) only by one step, it seems impossible to solve the conjecture (A) (or 1,1)) by some modifications of the present methods, even we cannot give a conditional proof by the assumption of (GRH) and the formula (33) with 8 = 1 — s, that is,

V(logx)cJ

which is usually called the Halberstam conjecture. Hence there are many who believe that Hardy's address that the conjecture (A) is "probably as difficult as any of the unsolved problems in mathematics" is still valid now as it was then. Hence it is convinced that a completely new idea is needed in the further study on Goldbach conjecture (A).

£ i • „ i _

max (h,k)=i

ir(x, k, h)

I. REPRESENTATION OF AN ODD NUMBER AS

THE SUM OF THREE PRIMES


SOME PROBLEMS OF 'PARTITIO NUMERORUM'; III: ON THE EXPRESSION OF A NUMBER AS A SUM OF PRIMES.

BY

G. H. HARDY and J. E. LITTLEWOOD. New College, Trinity College,

OxroiiD. CAMBRIDGE.

I . Introduction.

i . i . It was asserted by GOLDBACH, in a letter to EULER dated 7 June, 1742, that every even number 2m is the sum of two odd primes, and this proposition has generally been described as 'Goldbaeh's Theorem'. There is no reasonable doubt that the theorem is correct, and that the number of representations is large when m is large; but all attempts to obtain a proof have been completely unsuccessful. Indeed it has never been shown that every number (or every large number, any number, that is to say, from a certain point onwards) is the sum of 10 primes, or of 1 000000; and the problem was quite recently classified as among those 'beim gegenwartigen Stande der Wissenschaft unangreifbar'.1

In this memoir we attack the problem with the aid of our new transcendental method in 'additiver Zablentheorie'.' We do not solve it: we do not

1 E. LANDAU, 'Gel&ste und ungelOste 1'robleiue aus der Theorie dor Primzahlverteilung uiid der Riemannschen Zetafunktion', Proceedings of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. i, pp. 93—108 (p. 105). This address was reprinted in the Jahresbericht der Deutschen Math.-Vereinigung, vol. 21 (1912), pp. 208—228.

* We give here a complete list of memoirs concerned with the various applications of this method.

G. H. HAUDY.

/ . 'Asymptotic formulae in combinatory analysis', Comptes rendus dn quatrieme Congr'es des mathematiciens Scandinavcs a Stockholm, 1916, pp. 45—53.

2. 'On the expression of a number as the Bum of any number of squares, and in particular of rive or seven', Proceedings of the National Academy of Sciences, vol. 4 (1918), pp. 189—193.

1922, 3 (with J. E. Littlewood) Acta Malhematica, 44, 1-70.

2 G. H. Hardy and J. E. Littlewood.

even prove that any number is the sum of ioooooo primes. In order to prove anything, we have to assume the truth of an unproved hypothesis, and, even on this hypothesis, we are unable to prove Goldbach's Theorem itself. We show, however, that the problem is not 'unangreifbar', and bring it into contact with the recognized methods of the Analytic Theory of Numbers.

3. Some famous problems of the Theory of Numbers, and in particular Waring's Problem' (Oxford, Clarendon Press, 1920, pp. 1—34).

4. On the representation of a number as the sum of any number of squares, and in particular of five', Transactions of the American Mathematical Society, vol. 21 (1920), pp. 255—284.

j . Note on R a m a n u ^ ' s trigonometrical sum cq(n)', Proceedings of the Cambridge Philosophical Society, vol. 20 (1921), pp.263—271.

G. H. HARDY and J. E. LITTI.EWOOD.

/ . A new solution of Waring's Problem', Quarterly Journal of pure and applied mathematics, vol. 48 (1919), pp. 272—293.

2. 'Note on Messrs. Shah and Wilson's paper enti t led: On an empirical formula connected with Goldbach's Theorem', Proceedings of the Cambridge Philosophical Society, vol. 19 (>9'9), PP- 245—254-

3. 'Some problems of 'Partit io numerorum' ; I: A new solution of Waring's Problem', Nachrichten von der K. Gesellschaft der Wissenschaften zu Gbttingen (1920), pp. 33—54.

4. Some problems of 'Partit io numerorum' ; I I : Proof that any large number is the sum of a t most 21 biquadrates ' , Mathematische Zeitschrift, vol. 9 (1921), pp. 14—27.

G. H. HARDY and S. RAMANUJAN.

/ . 'Une formule asymptotique pour le nombre des parti t ions de n\ Comptes rendus de I'Acadimie des Sciences, 2 Jan. 1917.

2. 'Asymptotic formulae in combinatory analysis' . Proceedings of the London Mathematical Society, Ber. 2, vol. 17 (1918), pp. 75—115.

3. 'On the coefficients in the expansions of certain modular functions', Proceedings of the Royal Society of London (A), vol. 95 (1918), pp. 144—155.

E. LANDAU.

/ . 'Zur Hardy-Littlewood'schen JLOsung des Waringschen Problems' , Narhrirhlett von der K. Gesellschaft der Wissenschaften zu Gbttingen (1921), pp. 88—92.

L. J. MORDELI..

1. 'On the representations of numbers as the sum of an odd number of squares', Transactions of the Cambridge Philosophical Society, vol. 22 (1919), pp. 361—372.

A. OSTROWSKI.

/ . 'Bemerkungen zur Hardy-Littlewood'schen Ldsung des Waringschen Problems' , Mathematische Zeitschrift, vol. 9 (1921), pp. 28—34.

S. RAMANUJAN.

/ . 'On certain trigonometrical sums and their applications in the theory of numbers' , Transactions of the Cambridge Philosophical Society, vol. 22 (1918), pp. 259—276.

N. M. SHAH and B. M. WILSON.

/ . On an empirical formula connected with Goldbach's Theorem', Proceedings of the Cambridge Philosophical Society, vol. 19 (1919), pp. 238—244.

Partitio numcrorutu. Ill: On the expression of a number as a sum of primes. 3

Our main result may be stated as follows: if a certain hypothesis (a natural

generalisation of Riemann's hypothesis concerning the zeros of his Zeta-function)

is true, then every large odd number n is the sum of three odd primes; and the

number of representations is given asymptotically by

< • • « > ^ -ST^P^T) ' where p runs through all odd prime divisors of n, and

the product extending over all odd primes -us.

Hypothesis R.

i. 2. We proceed to explain more closely the nature of our hypothesis.

Suppose that q is a positive integer, and that

h = <p(q)

is the number of numbers less than q and prime to q. We denote by

x(n) = Xk(n) (k= i , 2, . . ., h)

one of the h Dirichlet's 'characters' to modulus q1: z, is the 'principal' character.

By i we denote the complex number conjugate to x- •/. is a character.

By L(s, %) we denote the function defined for a > i by

L{a)- L(a + it) - L{t, x) = Ms, Xk) - %*£• n- 1

Unless the contrary is stated the modulus is q. We write

L(s)*=L(s,x). By

Q = ft + iy

1 Our notation, so far as the theory of i-functions is concerned, is that of Landau's Handbuch der Lehre von der Verteiltmg der Primzahlen, vol. I, book 2, pp. 391 et seq., except that we use q for his k, k for his x, and sr for a typical prime instead of p. As regards the 'Farey dissection', we adhere to the notation of our papers J and 4.

We do not profess to give a complote summary of the relevant parts of the theory of the i-functions; but our references to Landau should be sufficient to enable a reader to find for himself everything that is wanted.

4 G. H. Hardy and J. K. Littlewood.

we denote a typical zero of L(s), those for which y = o, (i <=o being excluded.

We call these the non-trivial zeros. We write N(T) for the number of Q's of

L(s) for which o<y<T.

The natural extension of Riemann's hypothesis is

HYPOTHESIS R*. Every Q has its real part less than or equal to ~l

Wo shall not have to use the full force of this hypothesis. What we shall

in fact assume is

HYPOTHESIS R. There is a number &<^ such that 4

for every Q of every L(s).

The assumption of this hypothesis is fundamental in all our work; all the

results of the memoir, so far as they are novel, depend upon it *; and we shall not

repeat it in stating the conditions of our theorems.

We suppose that 0 has its smallest possible value. In any case ©>, —•

For, if Q is a complex zero of L(s), <j is one of L(s). Hence i — y is one of L(i — s), and so, by the functional equation 5 , one of L(s).

Further notation and terminology.

i . 3. We use the following notation throughout the memoir.

A is a positive absolute constant wherever it occurs, but not the same

constant at different occurrences. £ is a positive constant depending on the

single parameter r. O's refer to the limit process n — 00, the constants which

they involve being of the type B, and o's are uniform in all parameters except r.

•as is a prime, p (which will only occur in connection with n) is an odd

prime divisor of n. p is an integer. If g = i , p = o; otherwise

o<p<g, ( p , j ) = i .

(m,n) is the greatest common factor of m and n. By m n we mean that n is

divisible by m; by m n the contrary.

sl(n), n(n) have the meanings customary in the Theory of Numbers. Thus

sl{n) is log vf if n = -crm and zero otherwise: ^(71) is (—1)* if re is a product of

1 The hypothesis must be stated in this way because

(a) it has not been proved that no L(s) has real zeros between - and i,

(b) the Z-functions associated with tmj»'imiftve(uneigentlich) characters have zeros on the line 0 = 0. 1 Naturally many of the results stated incidentally do not depend upon the hypothesis. ' Landau, p. 489. All references to 'Landau' are to his Handbuch, unless the contrary is stated.

Partitio numerorum. Ill: On the expression of a number as a sum of primes. 5

k different prime factors, and zero otherwise. The fundamental function with which we are concerned is

(i. 31) /(*) = ^ log ^ *?• a

To simplify our formulae we write

e(x) = e2""'*, eq{x) = el-\-

Also

(1. 32) c , ( n ) = 2 M » P ) -p

If yik is primitive, q

(1. 33) »•* —r(z*)=- ^lfq{p)Xk(p) = 2leq(m)xk{m).' p m— 1

This sum has the absolute value1 \fq.

The Farey dissection.

1. 4. We denote by / ' the circle

(1. 41) \x\ = e~H=~e ».

We divide / ' into arcs fPl8 which we call Farey arcs, in the following manner. We form the Farey's series of order

(1.42) N-[fa~\,

the first and last terms being - and - • We suppose that is a term of the

series, and —, and %r the adjacent terms to the left and right, and denote by g r

ip.q ( ? > i ) t n e intervals

by 70,1 and /i,i the intervals lo, „ ^ I and l i— * >il- These intervals just

1 y.t(m) a° i f ( , H > ? ) > •• 1 Lanrinu, p. 497.


fill up the interval (o, i ) , and the length of each of the parts into which j ^ is

divided bv - is less than — and not less than — r . - If now the intervals j , . J q qN 2qN " *

are considered as intervals of variation of — . where 0 = a rgx , and the two in

extreme intervals joined into one, we obtain the desired dissection of r into arcs §Pl t . ' When we are studying the arc fP l j , we write

ipni (1.43) x = e « X-=eq(p)X=et(p)e-r,

(1.44) Y-tj + it).

The whole of our work turns on the behaviour of f(x) as |a;| — 1 , IJ — o, and

we shall suppose throughout that 0 <»?<,-• When x varies on §,,,,, X varies

on a congruent arc fp,,, and

* = - ( a r g * - ^ )

varies (in the inverse direction) over an interval — ^ p . q ^ ^ ^ p . v Plainly ttp,t

and 0'„ a are less than —5= and not less than —&> so that *•* qN qN

0,,q = Max (0p,q, (fPlt) < - ^ •

In all cases Y~' — (ij + %0)~' has its principal value

exp (—« log (i)+ •'«)),

wherein (since J; is positive)

— -rr<31og(i? + !0)<-7r .

By NT(n) we denote the number of representations of n by a sura of r primes, attention being paid to order, and repetitions of the same prime being allowed, so that

(1-45) 2i\Mn)z" = (2*0) r-

1 The distinction between major and minor arcs, fundamental in our work on Waring's Problem, does not arise here.

27


By vr(n) we denote the sum

(1.46) >v(n)= 2 log-nr, log-G7, . . . log-n;. cr, + a, + • - • + ov - "

so that

(1.47) 2 " ' W 1 " = ( / ( * » ' •

Finally <Sr is the singular series

(,48) ^ = S (^ ) r c ( -n ) .

2. Preliminary lemmas.

2. 1. Lemma 1. If »? = 9f(y) > o <Ae?i

(2. 11) /(*) = /,(*> + / , (* ) •

wAcre

(2. 12) /1(a;) = 2 ^ W : , : n - 2 l o g , n r ( a : a 2 + a : C T : ' + •••)• (J, n) > I ST

•J f I «

(2.13) f1(x)"^fy-r(s)Z{a)da, 2 —I »

y~ f /tas *te principal value,

. . . ' * * < « >

C* depends only on j), q and y.k.

(, I 5) c , — e £ l and

(2- 16) | C * | < ^


We have

= 2 M w> 2 ^ ('?+i)e_'"+Jl y

' 2-ioo

2 + ioo

2 - t «

where

• — . fy-'i'(*)2(«)ds,

*w-2*1*2^?-Since (<?,•;') = i , we have'

and so

where

* £'*(«)

Since jf*(;') = o if (9 , ; ' )> i , the condition (q,j)"i may be omitted or retained at our discretion.

Thus*

1 <_ m <_g, (g, wi)» I

1 Landau, p. 4*1. * Landau, pp 572—573.

Pnrtitio numerorum. Ill: On the expression of a number as a sum of primes. 9

Again, if k > i we h a v e '

C"- 1 l«f(P;)»(7) ^ J S ' f ( » ) » ( m ) . j—1 m— 1

If xu is a primitive character,

m - 1

If z i s imprimitive, it belongs to 9 = g> where d>i. Then **'w») has the

period Q, and

2 '» (™) **(»») = 2««(») z*(») 2 e«(/(?)-m —1 n — 1 / —0

The inner sum is zero. Hence C* — o, and the proof of the lemma is completed." 2. 2. Lemma 2. We have

(2. ai) I/. (*)!<;* dog (* + !))'• i f F.

We have

/.(») - 2 ^ ( M ) * " - 2 Iog « ( « 8 ' + * " " + • • • ) - / I , I ( * ) - / I . * ( * ) .

But

i/.,.(*)i-<=21°8^2i:i:rr

<A log(? + i)log52l :El2r<^( Iog(? + I)) ,2e~"2r

r-i r - 1

<^(log( ? +l))Îogy04( log( ? +l)) J S 2.

1 Landau, p. 485. The result is slated thero only for a primitive character, but the proof is valid also for an imprimitive character when (p,?)= 1.

* Landau, pp. 485, 489, 492. ' Seo the additional note at the end.


Also

ÎORHKAVS,

r>2,ra r2v'»i iBi" r^2 , a n

_ 1 _1_ <A{i — \x\) 2<Ar, 2 .

From these two results the lemma follows. 2. 3. Lemma 3. We have

(2.31) ^-TT = + + b <M + 2J - + - •

tuAere

<Ae o's, b's, b's and i's ore constants depending upon q and x, a is o or 1,

(2. 32) b, = i , b4 = o (fc>i),

and

(2. 33) o<s'b<A log (2 + 1).

All these results are classical except the last.1

The precise definition of b is rather complicated and does not concern us. We need only observe that b does not exceed the number of different primes that divide q,* and so satisfies (2. 33).

2. 41. Lemma 4. If o<i)< !-> then

(2-4") / ( * ) - ^ + 2C* f f* + P-

where

(2-412) O i - 2 r ( j ) 7 - » ,

1 Landau, pp. 509, 510, 519. ' Landau, p. 511 (footnote).


(2 .413) \P\<AV-q(\0g(g + 1))A£it\blc\+V-1*+\Ypd-k,

(2. 414) 8 = arc tan . - i*

We have, from (2. 13) and (2. 14),

2+100

(2. 415) U(x) = ^Ti fY-rit)Z[a)dt

2—'loo

2—too

say. But * t

2—too

where

*-{'-"«a ^/(a))o denoting generally the residue of /(«) for a — o.

Now*

•^'M 1— n • v t, log •CT'y , y i , log tSv

Zw = gQ £l~K=^ £«1— * 1 ... /»+<»l 1... / i — * + <i\ L'(i— s)

( 1 - * ) ' .^.i,^.*}!

where Q is the divisor of q to which % belongs, c is the number of primes which

divide q but not Q, tSltvi2, .. . are the primes in question, and e, is a root of

unity. Hence, if a =• . w e have

' This application of Cauchy's Theorem may be justified on the tines of the classical proof of the 'explicit formulae' for ^(x) and x(x): see Landau, pp. 333—3&>- I n t h i s c a B e t h e

proof is much easier, since Y~' /*(«) tends to aero, when |<| — *». • * • »n exponential <s~"' '. Compare pp. 134—135 of onr memoir 'Contributions to the theory of the Riemann Zetafunction and the theory of the distribution of primes', Acta Matktmatiea, »<*. 41 (»»t/T),. PP- "9 — '96-

' Landau, p. 517.

12

(2. 417)

G. II. Hardy and J. E. Littlewood.

I £ g j . I < A log q + A c log q + A log (| * | + 2) + A

< ^ ( l o g ( ? + i ) ^ l o g ( | < | + 2).

Again, if a = — +it, Y — rj + iO, we have

y—| = |y |<exp(«arc t a n ^ j .

\Y— r{s)\<A\Y\*(\t\ + 2) < e x p ( - ^ > r - a r c tan ®H <A\Y\ 4 Ml 2

log (|<| +2)" • inl

and so

(2. 418) l - i - , fY-T(s)^\ds < A (log (q + i))A \ Y \< fr^-e-t'dt \2TtlJ L\S) J

0

1 1

<A(\og(q + i))A\Y\*d ».

2. 42. We now consider R. Since

2 f c h + 3 - (—)• we have

n-{(t + 6)rw>. + P—!? y - r ( 4 - - ^ r

-j41(b + &)-(.b-b)UI + iMog 7) +0,(o)+ C,(o) log r ,

where each of the C's has one of two absolute constant values, according to value of n. Since

we have

(2. 421)

o ^ t < 1, o < b < 4 log (3 + 1), |log Y\<A log—<Ati -,

\R\<A\b\ + Alog(q+i)t) 2.


From (2. 415), (2. 416), (2. 418), (2. 421) and (2. 15) we deduce

\Pt\<A(log(q+i))*[\b\ + 1r* + \Y\*d"i),

(2.422) / , ( * ) - 1 ^ + 2 c * £ * + p . ft

(2.423) m<^<7(i°g(5+i)Wx2iM+f * + m V * ) .

Combining (2. 422) and (2. 423) with (2. 11) and (2. 21), we obtain the result of Lemma 4.

2. 5. Lemma 5. If q>i and y.k is a primitive {and therefore non-principal') character, then

— I •• M 2 where a = a{q, 2) = a*,

(2. 521) | L ( I ) | - * J 2|£(«')l ( « - i ) .

(2.522) I £ ( i ) | = 2 g 2 |L ' (o) | ( 0 - 0 ) .

.Fwr/Aer

(2.53) i - © O H ( ? ) < 0 ,

and

(2.54) l4^|<^(iog(3+i)^. 1 £,(1' 1

This lemma is merely a collection of results whicli will be used in the proof of Lemmas 6 and 7. They are of very unequal depth. The formula (2. 51) is classical.* The two next are immediate deductions from the functional equation for L(s).' The inequalities (2. 53) follow from the functional equation and the

1 Landau, p. 480. ' Landau, p. 507. 3 Landau, pp. 496, 497.


absence (for primitive %) of factors i — ty%f^' from L. Finally (2. 34) is due

to GRONWALL.'

2. 61. Lemma 6. If M(T) is the number of zeros Q of L(s) for which

o <,r <,lH<y +1 . then

(2. 611) M(T)<A(\og(q + i))^log(T+2).

The Q'B of an imprimitive L(t) are those of a certain primitive L(s) corres

ponding to modulus Q, where Q\q, together with the zeros (other than s = o)

of certain functions

E,= i — £*•&-',

where

|*, | — 1 , -cTvlg-

1 T. H. GnoNWAi.i., 'Sur les series de Dirichlet correapondant a des caractferes complexes', Rendiconti del Circolo Matematico di Palermo, vol. 35 (1913), pp. 145—159. Gronwall proves that

j - ^ | < 4 1 o g j(loglogj)8

fur every complex x, and states that the same is true for real x if hypothesis R (or a much less stringent hypothesis) is satisfied. LAXDAU (Uber die Klassenzahl imaginar-quadratischer Zahlkorper', Gottinger Nachrichten, 1918, pp. 285—295 (p. 286, f. n. 2)) has, however, observed that, in the case of a real x, Gronwall's argument leads only to the slightly less precise inequality

IT70I < A '° 8 q l / l o « l °S5 -

Landau also gives a proof (due to HECKE) that

fZIOl<ilu*«

for the special character ^—âssociated with the fundamental discriminant —q.

The first results in this direction are due to Landau himself ('ttber das Nichtverschwin-den der Dirichletschen Keihen, welche komplexen Charakteren entsprechen', Math. Annalen, vol. 70 (1911), pp. 69—78). Landau there proves that

jX(Tyj < A ( l08 «)B

for complex x-It is easily proved (see p. 75 of Landau's last quoted memoir) that

I/,'(i)|< A (log 9 ) \

so that any of these results gives us more than all that we require.


The number of •rav's is less than A Jog (9+1) , and each E,, has a set of zeros, on <j = o, at equal distances

> log sr, l o g ( y + i )

The contribution of these zeros to M(T) is therefore less than A (log (g + i))8; and we need consider only a primitive (and therefore, if q > i , non-principal) L{s).

We observe: (a) that a is the same for L(s) and L(s);

(b) that L{s) and L(s) are conjugate for real *, so that tho b corresponding to

L(s) is 6, the conjugate of the b of L(s); (c) that the typical e of L(s) may be taken to be either Q or (in virtue of the

functional equation) i — Q, SO that

«-2(^HrJ-2(j+j) is real.

Bearing these remarks in mind, suppose first that a = i . We have then, from (2. 51) and (2. 521),

L ( i ) L d ) Mo)Z(o)l

= A •n((.-;)«Vn((-^)|

Thus

(2.612) \2W(b) + S\<Alog{q+i).

On the other hand, if o = o, we have, from (2. 51) and (2. 522),

\L(i)Ui) q W(o)L'(o) •A"a(H)-'¥H-Hi'Ti'')l

and (2. 612) follows as before. 2. 62. Again, by (2. 31)

(2 . 621) JUj.i) i.»+,_;,(i±.) + 2ti_.g.


for every non-principal character (whether primitive or not). In particular, when •I is primitive, we have, by (2. 621), (2. 54), and (2. 33),

(2. 622) |»(*) + « 8 f | - | » ^ - b + | v ( ^ ) | < 4 ( l o g ( j + i))^.

Combining (2. 612) and (2. 622) we see that

(2. 623)

and

(2. 624)

S < ^ ( l o g ( g + i))^

|De(6) !<^( log( ? +i)H.

2. 63. If now g>i, and % is primitive (so that b = o), and a — s + iT, we have, by (2. 31), (2. 33), and (2. 624),

< . d + 4 1 o g (3 + 1) + ^ (log ( 5 + 1 ) ^ + 4 log ( |r | +2)

< ^ ( l o g ( 9 + i))Alog(|T|+2),

_2 l^^T-y)'<A ('°g (g + I ) H log ( l r | + 2)-

Every term on the left hand side is greater than A, and the number of terms is not less than M(T). Hence we obtain the result of the lemma. We have excluded the case 9 = 1, when the result is of course classical.1

2. 71. Lemma 7. We have

(2. 711) \b\<Aq(\og(q+i))A.

Suppose first that % is non-principal. Then, by (2. 621) and (2. 54),

(2.712) |6|<>1 (Iog(g+i))^+|2(7^-+f) |-

1 Landau, p. 337.


We write

(2-713) 2 = 2 , + 2 , '

where 2 ' s extended over the zeros for which 1 — ©^SRfg)^© and 2 o v e r

those for which 5R(p) = o. Now 2 —S'> where S' is the S corresponding to a

primitive L(s) for modulus Q, where Q\q. Hence, by (2. 623),

(2. 714) | 2 , | < ^ ( log(Q+i ) )^<^ l (Iog(</ + i ) )^ .

Again, the ?'s of 2 a r e *ê z e r o s (other than »-= o) of

the •ra'v's being divisors of q and ev an »n-th root of unity, where m=(p{Q)<qi; so that the number of tffv's is less than A log q and

where either wv = o or

-<K|<--q'-' '—2

Let us denote by QV a zero (other than « —o) of 1 — svts7', by ?'» a g, for which

|?v|<,i , and by Q", a q, for which | f v | > i . Then

(a . 7 I 5 ) |2(r^^)|^2(2+2)

Any gv is of the form

— + 1

« — e ?

2ni(m + uv) ?v ~ log 13;

where m is an integer. Hence the number of zeros q'v is less than A log -0, or than 4 log (q + i); and the absolute value of the corresponding term in our sum is less than

(2-7l6) m < n 2 r < i l ' l 0 8 ( g + 1 ) :

1 For (Landau, p. 48s) t — X(«rv), where X is a character to modulus Q.

18

so

( 2 .

that

7^7)

Also

( 2 . 718)

G. II. Hardy and J. E. Littlewood.

<-4 9 ( log(g + i))«

SI-ÎM—5 <2rr.

< A (log «;)» 2 ^ i < A (log (q + i))».

From (2. 715), (2. 717) and (2. 718) we deduce

(2.719) \22\<Aq{\og(q+ !))*;

and from (2. 713), (2. 714) and (2. 719) the result of the lemma. 2. 72. We have assumed that % is not a principal character: For the

principal character (mod. q) we have '

£,w=n(*-^»-cr| q

Since (1 = 0, b = i , we have

log T* £(£)_! / , («)

G7 9

b - i

©•—1 C(a) Z,,(s)

— I 2 \2 / • ^ \ « — » (fl

2 i 2 i^ + , i m (n4 + _L. ) = b _ I + 6_I^(l) + 2(- i - +I), *• or — i , _ i \ f ( a ) * — 1 / 2 \2/ A d \ i — Q Ql

\b\<A log (3 + 1) + 2fcVi) This corresponds to (2. 712), and from this point the proof proceeds as before.

1 Landau, p. 423.

' ^ | refers to U10 complex zeros of £,(»), not merely to those of CW

where

(a-

(2.

(2.

812)

813)

814)


2. 81. Lemma 8. If o < v < - then ' = 2

< 2 - 8 I I > /(,>-$+i<*fc+/>. A X * - i

\P\<AVqOog(q + l))A\<q + ^ * + \Y\*d 2J,

<J= arc tan j ^ - . -

This is an immediate corollary of Lemmas 4 and 7.

2. 82. Lemma g. If o <y<,~ then

(2. 821) /(*) = <? + ©,

where

(2. 822) V^'

(2. 823) | © | < ^ ^ ( l o g ( g + i))'4(<? + i / ~ ? + | y | - e < ' ~ 9 ~ T l o g ( | + 2 ) ) '

(2.824) <J = arc tan [-2.•

We have

(2 .825) 10*1 < 2 , ™ * - • ! + 2 J ' W y " ' l -

where i , extends over g*'s for which |y | ,> i , 2i2 o v e r t n o s e f ° r which | / | < i .

In i , we have

\I'(Q) Y-»\ - I/-(/» + »»11 7 h ' e x p (y arc tan £)

< A \rf-*\ y h ' exp (~ (5 « - arc tan ^ j |y|)

s - 1

< 4 M 2 | I ' f - e e - ' l ' l


(since | 7 | < A and, by hypothesis R, p<^@). The number M(T) of g's for which |y | lies between T and T + i(T>=o) is less than A (log {q + i))A log (T + 2), by (2. 611). Hence

2 |y|9 "e-'MÂ (log(q+j))Aî(n+i)e~~1\og(n + 2)e-i" n - 0

< ^ (log (g + 1))* d_e~"2 log (^ + 2^.

(2.826) 21lr(e>y_,'l<'4 Oogte + i ^ i r h 9 ^ ' ' ' ^ ^ 2 ) -

2. 83. Again, once more by (2. 6 u ) , 2 n a s a t moat A (l°g (? + *))•* terms.

We write

2 applying to zeros for which 1 — 0 <_$<_&, and 2 t 0 those for which ft = o.

Now, in 2 •

I Y-»\ - 1 Y|-/»exp iy arc tan ^) < ^ | Y\-f>;

and in 2 . | ^ ( ? ) l< -^ - Hence

(2 .832) | 2 2 j l | < ^ | y ^ 2 2 . l i r ( ? ) i < ^ i 7 i - 9 2 2 , 1i < ^ ( l o e ( 9 + i » ' 4 i y f " 8 -

Again, in 2 » > | l ' l < ^ and

^<Aq\og(q + l),

by (2. 716); so that

(-833) |2J<^2jn?)i^2j-^

<^22,2|7|<^«(log(?+I)^-

From (2. 825), (2. 826), {2. 831), (2. 832), and (2. 833), we obtain

(2- 834) \Gk\<A(log(q+i))*{q + \Y\-0d-e-t\og{±+2^-Hll,

Partitio uumerorum. Ill: Un the expression of a number as a sum of primes. 21

say; and from (2. 811), (2. 812), (2. 813), (2. 821), (2. 822) and (2. 834) we deduce

h

l®|- 2CkGk + P\

1 - 1

<^v^(iog(? + i)H(2+i?"»+|Ft-9^~8"*iog(J+2));

that is to say (2. 823). 2. 9. Lemma 10. We have

(2.91) h — <p{q)>Aq{\ogq)-A.

We have in fact '

for every positive 6", C being Euler's-constant.

3. Proof of the main theorems.

Approximation to vT(n) by the singular series.

3. 11. Theorem A. / / r »* on integer, ^ 3 , and

(3. in) C/Mr- 2 "'(">*"•

(3. 112) »v(») = 2 log«r, log «•,••• log «r„ Ufj+C. + . - . + STr-n

Men

(3. 113) Mn) = ^—^Sr + 0U <• <'(log n ) * J ~ ( 7 3 ^ 7 S r ,

1 Landau, p. 217.


where

(3. "4) * - 2 $ g r * < - » > -j - 1

It is to be understood, here and in all that follows, that O'a refer to the limit-process n — oo, and that their constants are functions of r alone.

If nj>2, we have

(3-1X5) *i*)-£nJM*)y£\'

the path of integration being the circle \x\ = e~H, where H = -< so that

! _ ! « ! _ I + 0 ( 1 ) 0 , 1 .

1 ' n \n*l n

Using the Farey dissection of order N •=[Vn'\, we have

(3.116) M«)-2 2 -nfuw^ » - l P < « . ( P . I > - 1 , •'

•=2e*(— nP)i*t> say. Now

\i'-<r\<\o\i\f'-i\+\i'-*<P\+-+\<p'-i\)

< .B (I ©/'-'l-Hay-11)-

Also I X~ n I — enH < A. Hence

(3- ii7) J M - ' M + W M >

where

(3- " 9 ) K , , l = O J j(\Of^\ + \<Dffr-l\)d0y

Partitio numerorum. HI: On the expression of a number as a sum of primes. 23

3. 12. We have rt=H = - and q<,Vn, and so, by (2. 823),

(3. 121) \<D\ < ArT* {log n)A + A (log n)AVq\Y\~e6~9~Tlog i^ + 2V

where (5 = arc tan Hh. We must now distinguish two cases. If |0|<,J7, we have

\Y\>AV, 5>A,

and

(3.122) Vq\Y\-^5 2log i + 2 \ < A n * r 1 - » = A n *.

If on the other hand 17 < |0|,< 0p,q, we have

S>A^>^\Y\>A\0\,

(3. 123) V g | y | - e r 9 " 2 1 o g ( j + 2 ) < ^ l / i . | ( 9 | - e . i 7 ~ e ~ 2 | « | e + 7 . 1 o g n

= ^ n 2 l o g n ( j | 0 | ) 2 < 4 f t M o g n . « *=*An Mogw,

since g|0|^.g0P,» <-<*B ? . Thus (3. 123) holds in either case. Also 0 ^ — and

so, by (3. 121),

(3. 124) | 0 » | < ^ n *(logn)A

3. 13. Now, remembering that r^^, we have

|v|I-'dfl<B&-«'- ,> M7|-<1-1>^

p,t " r,i

dO

<JSA-',-1'n'-2;

24 6. H. Hardy and J. E. Littlewood.

and so

(3-I3I) 2 [ \0<pr-1\dO<Bn'-*{M*x\O\) 2A _ ( r - 2 )

v. a y. a eP.t

< 5 » r _ 2 + e + 1Ml0g » ) B = 5 n r " ' + ^ 5(log „)B,

by (3. 124) and (2. 91). 3. 14. Again, if arg x — xp, we have

0

Similarly

;

>ff 2rt

\1\'d0=f\i -I'M °

00

= 2«:2(log •cr)M»l2ar< ^ 2 l o 8 m^(w»)|*P" tJ m—2

<-*d-i*i") 2 (2 lo« *-*(*)) i*P" m - 2 * - 2 '

oo

<.<4(i — |*|)2»»logm|a;|2m

m-2

<;r4n l 0g(r=y<^ l 0«n-

i/i<2lo«îa;r<2^(m)ia:im<7^]<^'1-

Hence

J,« 2 *

| / | ' -M®|<W<Max|©/ '- ' | M/|'«W _, »/

< B n 4 log n . n , - s . n log n

<JBnr-1 + (8_^(logn)*.

Partitio namerorom. Ill: On the expression of a number as a sum of primes. 25

From (3. 116), (3. 117), (3. 119), (3. 131) and (3. 141) we deduce

(3- 142) M«) = 2 M - » p ) ' M + 0 ( » ' ~ 1 + ^ ) ( l o g » H ' ) .

where l,,q is defined by (3. 118). 3. 15. In lp,t we write X = e~r, dX'= — e~TdY, so that Y varies on the

straight line from tj+iOp,, to r} — iffp,q. Then, by (2. 822) and (3. 118),

(3- 151)

1 + • » « > . Q

Now 1~'"'p,t !) + •'«

V+'l>p,q V—•'«

(3.152) - f = [Y-'enYdY + O(f\r!+i0\--do\ ' " t V—•'» 6q

- 2 *' (73^7 + ° [J\ 7 + i 01 -Tde) •

where

Also

Ot~Mm(0p,q,O'p,t)>-^-p<q zqw

GO 00

13- 153) f(t] + iO)-'dO< [&-'dO<BOt-r <B(qV^)r-\

Oq 6q

From (3. 151), (3. 152) and (3. 153), we deduce

(3-154) 2 M - » p ) ' p . , = ( ~ ^ 2 ( ^ | r M - » P ) + e . Pit

where

(3'I55) loK^SA-^-'n^-v^-^dr1

< Bnil"~n 2 (log ?)fl < Bn*' (log n)«.


Since r > 3 and 0 > - . -r<r— i <r—i+|© — -)> and from (3. 142), ==° = 2 2 4 " \ 4/ ^

(3. 154), and (3. 155) we obtain

(3- 156) n(«) - ^ 2 Q ' « . ( — P) + O ( n r - 1 + ( e -^( log »)*)

n - » V //j(g)\r , . , _ ( r - l + (e-{) _\ = [r~=^77 i W J « • ( - » ) + 0^» 4 /(logn)»;.

»<;JV

3. 16. In order to complete the proof of Theorem A, we have merely to show that the finite series in (3. 156) may be replaced by the infinite series <Sr. Now

n"~l 2 ( ^ M ) ' 0 * ( - n> I < Bn"~l 2 9'~T <lo8 9)B < Bnr (log n)*, 9>.tf r , : I " I , > a r

( © - - ) • Hence th is error may be absorbed in the second term

of (3. 156), and the proof of the theorem is completed.

and - r < r — 1 + 2

Summation oj the singular series.

3. 21. Lemma 11. If

(3- 211) c,(n) = 2 M » P ) .

where n is a positive integer and the summation extends over all positive values of p less than and prime to q, p — o being included when 9 = 1 , but not otherwise, then

(3. 212) c,(— n)=-cg(n);

(3. 213) <W(n) = c,(»)<v(n)

if (q, q') = 1; and

(3- 214) c,(n) = 2«J("(g)'

where <5 is a common divisor of q and n.

The terms in p and q — p are conjugate. Hence cg(n) is real. As c„(n) and cg{—n) are conjugate we obtain (3. 212).'

1 The argument fails if q •= 1 or q = 2; Imt c,()i) = c,(— n) = 1, f,(n) = t,(— n) — — 1.


Again

cq{n)ct, (n) = 2«cp(27171. ft + ^ ) ) = ^ e x p \ ~ p ] •

where

P-pq' + p'q.

When p assumes a set of (p(q) values, positive, prime to q, and incongruent to modulus 0, and p' a similar set of values for modulus q', then P assumes a set of <p{q)'p{q')="p{qq') values, plainly all positive, prime to qq' and incongruent to modulus qq'. Hence we obtain (3. 213).

Finally, it is plain that 1 - 1

d\q h-0

which is zero unless q\n and then equal to q. Hence, if we write

»/(?) = ? (?l»). i?(j) = o (?}»)> we have

d\q

a n d therefore

d\q

by the well-known inversion formula of Mobius.1 This is (3. 214).' 3. 22. Lemma 12. Suppose that r i>2 and

(3.-) * - 2 6gj)\<-«>.

(3. 222) Sr = 0

1 Landau, p. 577. ' The formula (3. 214) is proved by KAMANUJAN ('On certain trigonometrical sums and their

applications in the theory of numbers', Tram. Camb. Phil. Soc, vol. 22 (1918), pp. 259—276 (p. 260)). It had already been given for n = 1 by LANDAO {Handbuch (1909), p. 572: Landau refers to it as a known result), and in the general case by JKNSEN ('Kt nyt Udtryk for den talteoretiske Funk-

tion 2jp(n)~M(,n)\ Den 3. Skandinaviske Matematiker-Kongru, Kriitiania 1913, Kristiania (i9'5).

p. 145). Ramanujan makes a large number of very beautiful applications of the sums in ques

tion, and they may well be associated with his name.


if n and r are of opposite parity. But if n and r are of like parity then

(3 .2 2 3) &r-*Cr\l[ ( _ j ) r _ ^ 1 ^ — ) • p

where p is an odd prime divisor of n and

(3--4) 0 r _ n ( x - ^ ) . 13-3

Let

(3-«5) (jjgfc, (-»)-.<,. Then

fiqq') =i"(?)M(?') . 9>(<77') = <p(?)9>(g'). c„9 .(—n) — c„(—n)<v(—n)

'f (?> ?') = I> a n < i therefore (on the same hypothesis)

(3- 226) Aqq, = AqAq..

H e n c e '

Sr = A, + A, + A, + • • • = 1 + A2 + • • •=- U Xa

X3

where

(3- 227) xa — 1 + Aa + Atf +- Aaz + • • • = 1 + Aa,

since A0*, Aa*, ••• vanish in vir tue of the factor i>(q).

3. 23. If ta\n, we have

fi(ra,) = — 1 , <p(vt)-. T3—X, ca[n) = fi(&) = i ,

( 3 2 3 1 ) ^ — ( - l ) r

( t t f - i ) '

If on the other hand v/\n, we have

e«j (n) = n (•&) + vf(i (1) = vt — 1,

( - 1 ) ' (3- 232) Aa = ( • c r — i ) r - '

1 Since I fj(n) | i ^ j *, where i\n, we have CJ(M)= 0(i) when n is fixed and 17—00. Also

by Lemma 10, <f(q)> Aq(\ogq)~A. Hence the series and products concerned are absolutely convergent.

Partitio numerorum. HI: On the expression of a number as a sum of primes. 29

Hence

*-n("5£iMii(.-£3?)-W i n a j ii

If n is even and r is odd, the first factor vanishes in virtue of the factor for which tH = 2; if n is odd and r even, the second factor vanishes similarly. Thus Sr = o whenever n and r are of opposite parity.

If n and r are of like parity, the factor corresponding to •& •= 2 is in any case 2; and

9 , n L {~l)r \ n /(p-i)r+(-i>fQ>-i)\ S' - 2 II (* - (0—^) II 1 to-jy-i-v j •

0-3 » as stated in the lemma.

Proof of the final formulae.

3. 3. Theorem B. Suppose that r>=$. Then, if n and r are of unlike parity,

(3. 31) v,(n) =• o(nr~l).

But if n and r are of like parity then

2°r „ r_i n l(p-i)T + l-i)r(?-i)\ (3. 32) Mn) ~ j — ^ nr II (- ( p _ i r _ ( _ j ) r ) >

where p is an odd 'prime divisor of n and

Gt—3

This follows immediately from Theorem A and Lemma 12.' 3. 4. Lemma 13. If r ,> 3 and n anrf r ore 0/ Ziie parity, then

vt[n)> Bnr~\

for n>in,(r).

'Results equivalent to these are stated in equations (5. 11)—(5. 22) of our note 2, but incorrectly, a factor

(log n)~r

being omitted in each, owing to a momentary confusion between vr(n) and JVr(«i). The vr(«) of 2 is the Nr(n) of this memoir.

30 6. H. Hardy and J. E. Littlewood.

This lemma is required for the proof of Theorem C. If r is even

n(<-™^)>-If r is odd

€7— 3

In either case the conclusion follows from (3. 32). 3. 5. Theorem C. / / r >, 3 and n and r are of like parity, then

(3-51) NAn)co "'("> (log n)T

We observe first that

Nr(n)= ^ x ^ 2 1 < 5 n r ~ l

W , + CT,H hcr i .-»n mi + m j + - - + m r - n

and

( 3 - 5 " ) M11)— 2 log w , - - l o g w ^ ( l o g n) r2Vr(n)<Bn r-1 (log n)r

Write now

(3- 512) "r = *'r + A , # r = N'r + # " „

where v'r and iV'r include all terms of the summations for which

•cr.^M1-4 ( o < d < 1, » — 1 , 2 , . . . , r) .

Then plainly

(3. 513) *M«) > (1 - d)' (log n) ' tf ' r(n).

Again

N"r(n)<r% I 2 1 \ ra; <n 1 - 4 \wi+ <»•+' • •+av_i= » - «rr '

A ( » ) < (log nyN"r{n) Bnr~l for »:>»o(r)« by Lemma 13; and so

(3.514) (logn)'iV"r(n) = o(j'r(n)), v\(n) = o(r r(n)),

for every positive d.

Partitio numerorum. Ill: On the expression of a number as a sum of primes. ."1

From (3. 511), (3. 512), (3. 513), and (3. 514) we deduce

(i - 3 ) r (log n)r{Nr — N"r)<Vr— A £ ( l o g n)'Nr,

(1 —c5)r (log n)r2Vr!< vr + olvr)<{\ogn)rNr,

(i—d)r< Urn-.—~j.T> ilm -.—^-TTÎ-v = — (logn)rNr (log«)riv,—

As d is arbitrary, this proves (3. 51).

3. 6. Theorem D. Every large odd number n is the sum of three odd piimes. The asymptotic formula for the number of representations N,{n) is

(3. 61) j y , ( n ) o , C , ? r - g L - , n ( ( , > 7 l ) ( P ^ j ) ' l J ' J(logn)s-»M P* - 3 P + 3 /

where p is a prime divisor of n and

a-3

This is an almost immediate corollary of Theorems B and C. These theorems give the corresponding formula for N3(n). If not all the primes are odd, two must be 2 and n — 4 a prime. The number of such representations is one at most.

Theorem E. Every large even nvmber n is the sum of four odd primes (of which one may be assigned.) The asymptotic formula for the total number of representations is

where p is an odd prime divisor of n and

< 3 - 6 4 > O.-fUz-j^). 0 - 3

This is a corollary of the same two theorems. We have only to observe that the number of representations by four primes which nre not all odd is plainly O(n). There are evidently similar theorems for any greater value of r.

32 G. H. Hardy and J. E. I.ittlewood.

4. Remarks on 'Goldbach's Theorem'.

4. 1. Our method fails when r = 2. It does not fail in principle, for it leads to a definite result which appears to be correct; but we cannot overcome

the difficulties of the proof, even if we assume that ©>=•-• The best upper 1

bound that we can determine for the error is too large by (roughly) a powern*. The formula to which our method leads is contained in the following Conjecture A. Every large even number is the sum of two odd primes. The

asymptotic formula for the number of representatives is

(4. ID *,(.,«*a0l_i_Hij5i)

where p is an odd prime divisor of n, and

(4-") ^ = 1 1 l 1 - ^ ^ ) -We add a few words as to the history of this formula, and the empirical

evidence for its truth.' The first definite formulation of a result of this character appears to be

due to SYLVESTER*, who, in a short abstract published in the Proceedings of London Mathematical Society in 1871, suggested that

(4-3) tf.wco^ng^).

where 3,< •& < Vn, v8\n.

Since

n gE?)-n (*-vh?) n («-4H.n M « &<Vn « < V B a<Vn «<Yn

1 As regards the earlier history of 'Goldbach's Theorem', see L. E. DICISON, History of the Theory of Numbers, vol. 1 (Washington 1919), pp. 421—425.

' J. J. SYI.VKSTER, 'On the partition of an even number into two primes', Proe. London Math. Soe., ser. 1, vol. 4 (1871X PP- 4—6 (Math. Papeti, vol. 2, pp. 709—711). See also 'On the Goldbach-Enler Theorem regarding prime numbers', Nature, vol. 55 (1896—7), pp. 196—197, 269 {Math. Papon, vol. 4, pp. 734—737)-

We owe our knowledge of Sylvester's notes on the subject to Mr. B. M. WILSON of Trinity College, Cambridge. See, in connection with all that follows, Shah and Wilson, 1, and Hnrdy and Littlewood, 2.


and*

a<Vn

where C is Euler's constant, (4. 13) is equivalent to

(4 .x 3 ) N,{n)„4rc0tJ£_JL^.

and contradicts (4. 11), the two formulae differing by a factor 2 e - c = 1.123 . . . We prove in 4. 2 that (4. 11) is the only formula of the kind that can possibly be correct, so that Sylvester's formula is erroneous. But Sylvester was the first to identify the factor

« • * > n( )-to which the irregularities of N,(n) are due. There is no sufficient evidence to show how he was led to his result.

A quite different formula was suggested by STACKEL* in 1896, viz.,

, v ' (logn)»1Mp— 1/

This formula does not introduce the factor (4. 16), and does not give anything like so good an approximation to the facts; it was in any case shown to be incorrect by LANDAU ' in 1900.

In 1915 there appeared an uncompleted essay on Goldbach's Theorem by MERLIN.4 MERLIN does not give a complete asymptotic formula, but recognises (like Sylvester before him) the importance of the factor (4. 16).

About the same time the problem was attacked by BBUN'. The formula to which Brun's argument naturally leads is

' Landau, p. 218. ' P. STACKKI, 'Ober Goldbach's empirisches Theorem: Jedo grade Zahl kann ale Sumine

von zwei Primzahlen dargestellt werden', Gbttingtr Nachrichtm, 1896, pp. 292—299. ' E. LANDAU, 'Ober die zahlentheoretische Funktion e(n) und ill re Beziehung zuin Goltl-

bachschen Satz', Gottinger Nachrichten, 1900, pp. 177—186. ' J. MERLIN, 'Un travail sur les nombres premiers', Bulletin ties science! mathimaliquct,

vol. 39 (>9'5), PP- 121—136. • V. BRUN, 'Ober daa Goldbachsche Gesetz und die Anzahl der Primznhlpaare', Archiv for

Mathematik (Christiania), vol. 34, part 2 (1915X no. 8, pp. 1—15. The formula (4. 18) is not actually formulated by Brun: see the discussion by Shah and Wilson, 1, and Hardy and Littlewood, 2. Bee also a second paper by the same author, 'Sur les nombres premiers de la forme ap + b', ibid., part. 4 (1917). no. 14, pp. 1—9; and the postscript to this memoir.


(4.i7) ff.(»)~2ff»n(553'

where

(4. I7D fl_n j x - i ) .

This is easily shown to be equivalent to

(4.18) ^ ( n ) ^ 8 e - J ? C j _ j _ n ( ^ ) ,

and differs from (4. 11) by a factor A,e~ic = 1 . 263 . . . The argument of 4. 2 will show that this formula, like Sylvester's, is incorrect.

Finally, in 1916 STACKEL1 returned to the subject in a series of memoirs published in the Sitzvngsberichte der Heidelberger Akademie der Wissenschajten, which we have until very recently been unable to consult. Some further remarks concerning these memoirs will be found in our final postscript.

4. 2. We proceed to justify our assertion that the formulae (4. 15) and (4. 18) cannot be correct.

Theorem P. Suppose it to be true that1

( log w)» •"•!»> — 2/

n = 2 0 p" p ' ° ' . . . (a>o,a, a', ... > 0),

C7—3

' P. STACKEL, 'Die Darstellung der geraden Zahlen als Summen von zwei l'rimzahlen', 8 August 1916; 'Die LQckenzahlen rter Stufe und die Darstellung der geraden Zahlen als Sum-men und Differenzen ungerader Primzahlen', I. Teil 27 Dezember 1917, II. Teil 19 Januar 1918, III. Teil 19 Juli 1918.

* Throughout 4 . 2 A is the same constant.

(4. 21)

•7

and

(4. 22)

if n is odd.

(4- 23)

Then

Partitio numerorum. Ill: On the expression of a number as a sum of primes.

Write

(4. 24) fl(n)='4nn(Ê^) (" even1> fi(») = ° (» odd).

Then, by (4. 21) and Theorem C, now valid in virtue of (4. 21),

(4. 25) vAn)~* 2 l oS ^ l o8 •"' °° flW.

it being understood that, when n is odd,- this formula means

v1(n) = o(n).

Further let

,£i(n) ^n(n)

these series being absolutely convergent if 9f(«)>2, 9 t (u )> i . Then

(4.26) /w-^2»-nK53 n=0(m«1.2) » ^

- ^ y ,-<.» „-..« „.—-« Q ) - i ) ( p ' - i ) . . .

2 - M Vr-/ , nf—i c r » \ 2 - " ^ , , ,

say: Suppose now that u — 1, and let

Then

-n((«+^-J/(«+^))-n(J^)

**-\(vt-iy-i) c2

Hence

(4.27) / W ~ ^ w ~ ^ , ( « ) ~ A ? ( M ) ^ a g ^ _ f ) = 2Cfs_2)-


On the other hand, when x— i ,

and s o '

2 I O 3 ".

2 (4. 28) Vj(i) + v2(2) -\ t- v,(n)co -n*.

It is an elementary deduction' that

1 v,(n) a Is) •= >, ———- 00 >, 00

-2

•when a — 2 ; and hence ' that (under the hypotheses (4. 21) and (4. 22))

1 (4. 29) /(a)<>3

s — 2

Comparing (4. 27) and (4. 29), we obtain the result of the theorem.

4. 3. The fact that both Sylvester's and Brun's formulae contain an

erroneous constant factor, and that this factor is in each case a simple function

of the number e~c, is not so remarkable as it may seem.

In the first place we observe that any formula in the theory of primes,

deduced from considerations of probability, is likely to be erroneous in just this

way. Consider, for example, the problem 'what is the chance that a large number

n should be prime?' We know that the answer is that the chance is approxim

ately j J l o g n

Now the chance that n should not be divisible by any prime less than a

fixed number x is asymptotically equivalent to

» . ( - # sr<x

' We here use Theorem 8 of our paper Tauberinn theorems concerning power series und Dirichlet's series whose coefficients are positive', ftoc. London Math. Soc, ser. 2, vol. 13, pp. >74—'91- This is the quickest proof, but by no means the most elementary. The formula (4. 28) is equivalent to the formula

2 *,<»)<: 2 (log n)'

used by Landau in his note quoted on p. 33. ' For general theorems including those used here as very special cases, see K. KHOPT,

'Divergenzcharactere gewisser Dirichlet'scher Reihen', Acta Mathcmatica, vol. 34, 1909, pp. 165— 304 (e. g. Satz III, p. 176).


and it would be natural to inferJ that the chance required is asymptotically

equivalent to

sr<v>> But*

TT ( , _ . ! ) ^ « ^ ; **• \ -al log n

a> Vn

and our inference is incorrect, to the extent of a factor ze~c. It is true that Brun's argument is not stated in terms of probabilities',

but it involves a heuristic passage to the limit of exactly the same character as that in the argument we have just quoted. Brun finds first (by an ingenious use of the 'sieve of Eratosthenes') an asymptotic formula for the number of representations of n as the sum of two numbers, neither divisible by any fixed number of primes. This formula is correct and the proof valid. So is the first stage in the argument above; it rests on an enumeration of cases, and all reference to 'probability'1 is easily eliminated. It is in the passage to the limit that error is introduced, and the nature of the error is the same in one case as in the other.

4. 4. SHAH and WILSON have tested Conjecture A extensively by comparison with the empirical data collected by CANTOR, ADBRY, HAUSSNER, and RIPERT.

We reprint their table of results; but some preliminary remarks are required. In the first place it is essential, in a numerical test, to work with a formula N2{n), such as (4. 11), and not with one for v7(n), such as (4. 25). In our analysis, on the other hand, it is v,(n) which presents itself first, and the formula for N2(n) is secondary. In order to derive the asymptotic formula for N2{n), we write

vt{n) = 2 log •n' log •uf 00 (log n)' N2(n).

The factor (log n) ' is certainly in error to an order log n, and it is more natural5

to replace v2{n) by ((logn)' — 2 log n + •••)#,(»).

1 One might well replace a<VTi by a<n, in which case we should obtain a probability half as large. This remark is in itself enough to show the unsatisfactory character of the argument.

1 Landau, p. 218. • Whether Sylvester's argument was or was not we have no direct means of judging. ' Probability is not a notion of pure mathematics, but of philosophy or physics. 4 Compare Shah and Wilson, /. c, p. 238. The same conclusion may be arrived at in

other ways.


Por the asymptotic formula, naturally, it is indifferent which substitution we adopt. But, for purposes of verification within the limits of calculation, it is by no means indifferent, for the term in log n is by no means of negligible importance; and it will be found that is makes a vital difference in the plausibility of the results. Bearing these considerations in mind, Shah and Wilson worked, not with the formula (4. 11), but with the modified formula

N,(n) 00o{n) = 2C lT. rr^—j Tf i—1) • 2V ' VK ' 2(logn) '— 2 log n-**-\p —2/

Failure to make allowances of this kind has been responsible for a good deal of misapprehension in the past. Thus (as is pointed out by Shah and Wilson') Sylvester's erroneous formula gives, for values of n within the limits of Table I, decidedly belter results than those obtained from the vnmodified formula (4. 11).

There is another point of less importance. The function which presents itself most naturally in our analysis is not

/(*) = 2 loS **" but

g{x)=- 2 ^ ( » ) * " = ^ l o g t f r X .

The corresponding numerical functions are not v,{n) and N,{n), but

g,(n) = ^yi(m) yt(m'), Q,{n) -= 2 x

(so that Q2(n) is the number of decompositions of n into two primes or two powers of primes'). Here again, N,{n) and Q,{n) are asymptotically equivalent; the difference between them is indeed of lower order than errors which we are neglecting in any case; but there is something to be said for taking the latter as the basis for comparison, when (as is inevitable) the values of n are not very large.

In the table the decompositions into primes, and powers of primes, are reckoned separately; but it is the total which is compared with o(n). The value of the constant 2C3 is 1 . 3203. I t will be seen that the correspondence between the calculated and actual values is excellent.

1 /. c, p. 242.

Partitio numerorum. Il l : On the expression of a number as a sum of primes.

Table I.

H

3o = 2 . 3 5

32 = 2*

34 = 2.17

36 = ='.3'

210=2.3.5.7

2 1 4 = 2 . I07

216 = 23. 31

256 = 2"

2,048 = 2"

2,250= 2 .3*. 5s

2,304 - 2" . 3'

2,306 = 2. II53

2,3IO = 2.3.5.7.11

3,888 = 24.3'

3,898 = 2 . 1949

3,990= 2.3.5.7. 19

4,096 = 212

4,996= 21. 1249

4,998 = 2. 3.;', 17

5,000 = 2!.54

8,190 = 2.3'. 5. 7 .13

8,192 = 2"

8,194 = 2.17. 241

10,008 = 2*. 35. 139

10,010 =2.5.7.11.13

10,014 = 2.3. 1669

30,030=2.3.5.7.11.13

36,960 = 2''. 3 . 5 . 7 . 11

39,270= 2. 3. 5. 7.11.17

41,580= 2'. 3". 5. 7.11

50,026 = 2 . 25013

50,144 = 2'. 1567

170,166 = 2 3- 79-359

170,170 = 2.5.7.11.13.17

170,172 = 2'. 3'. 29. 163

«»(«)

6 + 4 =10

4 + 7 = " 7 + 6 =13

8 + 8 =16

42 + 0 = 4 2

17 + 0 = 1 7

28 + 0 = 2 8

16 + 3 = 1 9

5 0 + 17 = 67

174 + 26 = 200

134 + 8 = 142

67 + 20 = 87

228 + 16 = 244

186 + 24 = 210

99 + 6 = 105

328 + 20 » 348

104 + 5 = I09

124 + 16 = 140

228 + 20 = 308

150 + 26 = 176

578 + 26 = 604

150 + 32 = 182

192 + 10 = 202

388 + 30 = 418

384 + 36 = 420

408 + 8 = 416

1,800 + 54 = 1854

•>956 + 38 = 1994

2,152 + 36 = 2188

2,140 + 44 = 2184

702 + 8 = 7 1 0

60; + 32 = 706

3,734 + 46 = 378o

3,784 + 8 = 3792

3,732 + 48 = 3780

p{n)

22

8

9

17

49 16

32

•7

63 179

•36

69

244

'97

99

342

102

119

305

•57

597

171

219

396 384 396

'795 '937 2213

2125

692

694

3762

384'

3866

QM-PM

0.45

1.38

1.44

0.94

0.85

1.07

0.88

1 . to

I .06

I . 11

1.04

I . 26

I . 00

I . 06

1.06

1.02

1.06

1.18

I .01

1 12

I .01

I .06

O . 92

I . 06

1 .09

i.o5

' 03

' 03

0.99

'•°3

I.03

1 .02

1 .00

O.99

O.98

60

Noted by the editor: We omit the remaining part of the paper, where

a lot of conjectures on the other additive problems are given by

the circle method on considering only the major arcs.

Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS 1937. Volume XV, JVi 6—7

MATHEMATICS

REPRESENTATION OF AN ODD NUMBER AS A SUM OF THREE PRIMES

By I. M. VINOGRADOV, Member of the Academy

Some simple examples of application of my method to the theory of primes were given in 1934 J1).

In the present paper I give the application of the same method to the estimation of the sum

gSKiap

JXW

and with help of this estimation and using a new theorem concerning the distribution of primes in an arithmetical progression (2) (the difference of the progression grows slowly simultaneously with the increasing of the number of terms), I deduce an asymptotical formula for the number of representations of an odd number N > 0 in the form

N = Pi+ P2+Pa-It follows directly that every odd number from a certain point onwards can be represented as a sum of three primes. This is the complete resolution of Goldbach's problem for odd numbers.

The estimations of the present note can be replaced by much more exact estimations.

N o t a t i o n s . JV > 0 is an odd and sufficiently great number; n = = logJV.

h, hu ht, ... are arbitrary large constants > 3;

T = Nn-3*; TX = Nn ~h;

8 is a real number and | 0 | < 1 ;

A€B; A = 0(B)

denotes that the ratio -*-=•* does not exceed a certain constant; D

(d) denotes a set of divisors d < N of the product H of all primes < ^]/N; {d0) denotes the part of this set consisting of all d with an even number of prime divisors; (dx) is the part of the same set consisting of all d with an odd number of prime divisors. The set (d) is

2

divided also into two sets (d ) and (d ) . The first consists of the numbers d satisfying the condition that all prime divisors are

« n 3 h ,

the second consists of all the remaining numbers d. The sets [d0) and (d t) are correspondingly divided into sets (do), (do")

and(d'1),(d'1 '). Lemma 1. Let (x) and (y) denote two sets of increasing positive integral numbers;

l<U0<Ul<Nl<N; m is integral and > 0.

a = — + — la, q) = 1; 0 < q < x; m = m^; ? = ?1S; S = (m, g);

* W

where x runs over the numbers (x) satisfying the condition

u0 < x < r/x

and y for a given x runs over the numbers (y) satisfying the condition

° < y ~ x

Then we have

Theorem 1. Let

<* = f + £;(«,?)=i; n s h<?<T. Then we have

Proof. We have

d

S='^lV.(d)Sd + 0(|/JV); 5„ = 2e*""""1- (1) (<J) m - l

Hence we find S='^l>.(d)Sd + 0(Nn-^) =

d>*i

= Tt-Tt + 0[Nn-^)i r0 = 2 ^ ; ^ = 2 ^ , (2) «t.) (<*,)

where d runs over the values > T^ We estimate only T0 as the estimation of 7 \ is performed in the same way.

By change of the order of the summation we get

T0 = T(m); T (m) = J *«"*, (3) m d

where m runs over the numbers

m = l , . . . , [n»]

and d for every given m runs over the numbers

1 m Further we find

T(my=T'(m) + 0(£.n-») (4)

where T"(m) consists only of those terms of the sum T(m), which correspond to the values of d from the set (do). For, the part T' (m) of T(m) corresponding to the values of d from d0 does not exceed the number of those terms of the set (d') which

iV are not exceeding — . But the order of this number is significantly less than

m

— n-h. m

Now if d belongs to (d0') and k is the number of the prime divisors of d exceeding n 3 \ then

k<n. Therefore

7"(m) = 2 7 \ ( » ) , <5)

where Tk(m) consists of the terms of T"(m) with d containing exactly k prime divisors > n3h. Further we find

Th (m) = TT^(m) + ° d »"*) • (6> where

u o

and u runs over the primes > n3h belonging to (d) and v for a given u runs over the numbers belonging to (dt) and satisfying the condition

— < v< — • u mu

To the sum Th0 (m) we can directly apply the lemma 1. We obtain s

y m

and then from (6), (5), (4), (3) and (2) we deduce the theorem 1. Theorem 2. For the number IN of the representations of N in the

form N = Pi + p, + pa

we have the formula IN=>RS + 0{lfn-)t

where c is an arbitrary large constant > 3 and

5 = 2 ^ 2 ^ " ; * = £(!+*); limA = 0; f l-9(f). 9 - 1 * ' 0<a<» "-"*'

(a, « ) - l

* * h

Proof, a) We nave i

JN = f SI e-**i"N do.; Sa = 2 «*"'"*' •

The interval of integration we divide into intervals of two classes

2. The remaining intervals; for them

«=j+z; (a,g) = U n*»<q<-r, | * | < £ .

Correspondingly this division into intervals we have

IN = INI + IN*- 0)

b) Applying the theorem 1 we find I t

Im 4 yv»*~h f 15. |a da € Nn2~h f 2 2 e!*"(p_p,) <*<* ^

€Nn?-"~€N*n1-h. (8)

c) The calculation of ijyi we perform without difficulty. This is done in the same way as in the Waring's problem. But here we use the now theorem on the distribution of primes in an arithmetical progression. If a belongs to the interval of the first class then we find

N

^^m.V(z) + 0(Nn-">); V{z)=$^-dx.

Hence the part of IN1 corresponding the given fraction — is represented in the form q

R ^ e * • +0 (Nn-»>); R = f [ V (z)]s e-«"*«» dz,

where R can be expressed in the form

R = ^(i+X); HmA = 0.

Hence we find without difficulty:

Im = RS + 0(N*n->>*)

and by (7) and (8) the theorem follows directly.

V. Stekloff Institute of Mathematics. D . . Academy of Sciences of the USSR. "®SeJ™?

MOSCOW. 19.V.1937.

REFERENCES

» H. M. B H H o r p a n o B , flAH, III, 1; IV, 4(1934). » A. W a l f i t z , Math. ZS., 40 (1936).

A New Proof of the Goldbach -Vinogradov Theorem

Ju. V. Linnik

§1. In my note 'On the possibility of a method for some

'additive' and 'distributive' problems in the theory of prime

numbers" [l], I sketch a proof of the Goldbach problem by the pure

Riemann-Hadamard's method of L-series and contour integration with

some theorems on the density of zeros of L-series.

In this paper, I gave the detailed proof of the three primes

theorem by the Riemann-Hadamard's method, thereby the Hardy-

Littlewood's conditional solution is completed.

§2. Our basic instrument is the following lemma, and its

detailed proof is contained in my paper "On the density of the zeros

of L-series" [3] .

Fundamental lemma. Let q be a natural number, x(n) a primi

tive character (mod q) and L(u, x) its L-series. Let

to = a+ it , T > q , 6 > 1 and v = B - - > 0 .

Then the number of zeros of L(w, x) in the rectangle 6 < a < l ,

|t| <T will be

Q(B. T ) < c 1 q2 V - ^ 1 - V ) l n 1 0 T + c2q

30 , (1)

where c, and c2 are absolute constants.

§3. Let

S(N, 6) = I A(n) e-<n/N) e " 2 ™ 8

n=l

where N is an odd number which we wish to decompose as the sum of

three prime numbers. Then

Q(N) = e f S(N, e ) 3 e27rlN6 de + o( J0

N3/2 + E)

where

Q(N) = I Inplnp' lnp" p+ p' + p" = N

Let i ., 10,000 u 100 r = In N , x = r ' , H . = T

For each e e [0,1], we use the approximation by continued

fraction

6 = ^ + a , |a| < ^ , q < T (2)

The set JL formed by those 8 with |a| < H-.N- = T'U" N~ TOO „-I

ise B w i u n |a| < n-.iv = -

is called the "major arcs" |2| The asymptotic behaviour of S(N, 6) for 8eJt is well

established by the classical Riemann-Hadamard's method with the

aid of Siegel's theorem [4] or Page's theorem [5]. Hence the

integral

f S(N, 9) 3 e 2* i N e de J Jib

forms the principal term of our problem.

We use m to denote the complement of JH with respect to

[0,1]. For 6 e m, we have

e = | + a . J ^ l a l * ^ . q < T . (3)

Now we shall prove that for 6 € m, S(N, 6) can also be

estimated by the Riemann-Hadamard's method.

§4. We use x to denote a primitive character (mod q), where

q satisfies (3); E(x)= 1 if x is the principal character,

E(x)=0 otherwise, and p denotes a critical zero of L(u, x)-

Suppose that x is a number with Rex > 0. It follows by

Littlewood's argument [6] that if L(0, x) 0» then

S(N, a, X) = I x(n) A(n) e n=l

-nx

= E(x)x-] - I x"p r(P) -t' (0, x) p L

r-i + 1

2iri

2 x-"(-k'(., x))r(o i-i» v L ;

)du

and it remains no essential alternation if L(0, x) = 0-

Let x=N" + 2iTia, where a satisfies (3). Then |x| <

In order to estimate the remainder term

R = - M x"W (- f (">, X)j r(a))do. ,

we notice that for a= — , 2

L' , \ „ i „ / I J - I . O N -ID -tolnlxl- iwarc X - (a), x) « Inq ( | t | +2) , x = e ' '

-u ln lx l < 1 ,

r(«o)« I t f 1 e - ^ 2 ) ! *

- luarc x < e t arc x

TT 1 Take n = — arc x = arc tg 2 2-nNa

Then

R «

For e e m,

f e ( a r c x - W 2 ) ) t . 2 ^ d t « l r )3 1

i i 3 a r c t q — — > —'— , R « ( In Na) y 2irNa 4TrNa ^

i t _1 _] Since - (0, x) K< 9 a n d x « a , we have

S(N, a, X) « a"1 + 0 " Na)3 + I x"p r(P) p

§ 5 . Let v n = i l ™ 0 In N

In order to estimate |£ x~p r (p ) I , we

divide the c r i t i c a l s t r i p a into the sum of s t r ips a0-. 0 < a s - +

vQ = 6 0 , and those a p : 6 < a < 0 + y ^ p , 2 °

Let a>0

and N n>Na . It is well-known [5] that the trivial estimation for

the number of zeros of L(w, x) in the rectangle 0 < a < 6 , |t|<N„

will be

QL(B, NQ) « NQ ln(qN0)

Set p. = g. + i t k - Since | X | ~ 2 T O and

we have

| r (B+i t ) | < c^-t6-5 e-(^/2) | t |> | t | > T , - l < a < l

3 2

I * K r (p k )

0 p k e a p

(5)

-3 0 (arcx-(7y/2))|tk | B - J « a I e \t.\

Pke a6, 0

-30 -( | tk | /4^Na) eQ-l « a I e | t k |

p k e a e n

p0 0 J"vo „2vo « a Na ln(Na)(Na) « a N N ,

where the constant imp l i c i t in symbol « is uniform on x> N, a, v.

1 Since a =

q-r

'0 « ( In N)

we have

20,000 v 0 2 v0 2 2 « 1 , N u = ( In N)* = r ,

and therefore

| - v o 2 vo M " M Nr N N N « < (qT)i ' q M

(6)

§6. Suppose that -+v 0<6<0.6, i.e. vQ<v<0.1. Then by

the fundamental lemma (inequality (1)), we have

QL(P. NQ)< c^Nj-^-^l^Vô3 0

Notice that NQ> Na> H] = T 1 0 0 > q 1 0 0 . Hence

-Pi 1 x K r(P . )

p k e a B « a z ' I e

•(|tk | /4Trlfa) *" V T a ^ K " ' l t ' V

p k e a 3

« ( N a j T - W l - v ) q 2v ( N a ) v a - J -v ^ 1 0 ,

~ /M J - v 2v -J-v 10 « (Na) q a z r

2v 10 j l - v J - v - v M O n2v < Nq'v r

(qx)

N a ' r ~ q ' < - ^ : — ^ « J-v-v qt?

(7)

1 3 1 §7. Suppose that 0.1 < v < - . Then =- - -— > 0, and so

I x"Pk r(p. ) « of*"" (Na) 1-^ 7 1"^ (Na)vq2v ln10N P k e a 6

« N' v a

Since q< T, we have

l V J3/2)-(Vl-v) q2v K< Nl-0.01 q2v

I x K r(P,) pk e a6

qi N0-005 (8)

Suppose that -<v<0.4. Then ^ - - — < 0 , and the maximum

"V Hl estimation is obtained by the minimum of a>-rr-, i.e., the right-hand side of (8) is

70

<<Nl-(Al-v)r10q2v/A^(1/1-V)-<3/2)

(*)

N2+v 2v 10 o 9 10 N « — 9 r < Nu.y qriu < N ,g) H(Vl-v)-(3/2) ql N0.05 1

Finally, for 0.4 < v < - , the sum is dominated by

a"i-V ( N a ) 1 ' ^ 1 - ^ r10 q2v (Na)v

< N l - ( v2 / l - v ) r 1 0 q 2 v / i L \

( 1 / 1 - V ) - ( 3 / 2 )

<K N^W° K< _ N _ . (10)

H(5/3) - (3/2) q H 0 0.1

§8. For 6 = | + a (9em), . i t fol lows by ( 4 ) , . . . , ( 1 0 ) that

I X(n) A(n) e-(" /N) e ^ i n a

n=l

« r l l a f 1 + (In Na)3 + - j \ - + N

qiTi q j N 0 - 0 0 5

N

qi N°- 0 5 qH° 0.1

|a| « — - ,

qHf

and thus there is a small number c 5> 0 such that

I X(n) A(n) e - ( " / N ) e " 2 ™ 0 ' « - — » . ( I V n=l 5 + c5 0.1

q T

Consequently, for 9 =-|-+a (9 em), we have

S(N, 6) = I A(n) e " ( n / N ) e " 2 ^ ' " 6

n=l

= I A(n) e " ( n / N ) e2ir i(a/q + a)n

n=l

I e-2ir1(a/q)A j A ( n ) e~(n/N) e"2Trina

(A.q) = 1 nH£(mod q) £(modq)

+ 0 ( q e )

Since

I X (n) A(n) e " ( n / N ) e~Zirina + o(qe) n=l

we have by (11),

s(N, e) « aliiai l l « 4U < N

*(q) qi + C5 T0.1 TO- 1 ( I n N ) 1 ' 0 0 0

(12)

I t is su f f i c i en t fo r the solut ion of Goldbach problem.

References

[ l j . Ju. V. L innik , Dokl. Akad. Nauk SSSR, 48, (1945) 3-7.

[2]. E. Landau, Vorlesungen liber Zahlentheorie, Bd. II (1927).

[3]. Ju. V. Linnik, Nzv. Akad. Nauk SSSR, Ser. Mat; 10 (1946) 35-46.

[4]. C. L. Siege!, Acta Arith; 1 (1935) 83-86.

[5]. A. Page, Proc. London Math. Soc; 39 (1935) 116-141.

[6]. J. E. Littlewood, Proc. London Math. Soc; 27 (1928) 358-371.

(See Mat. Sbornik, 19 (1946) 3-8).

Translated by Wang Yuan

A N e w Proof on the Three Primes Theorem

Pan Cheng Biao

(A) By the circle method of Hardy and Littlewood and his method

on the estimation of trigonometrical sum with prime variables, I. M.

Vinogradov [l] first proved in 1937 that every large odd number is

the sum of three prime numbers which is usually called the Goldbach-

Vinogradov theorem or the three primes theorem. Later, Ju. V.

Linnik [2] and N. G. Tchudakov [3] gave another two proofs on this

theorem based on the estimation of the density of zeros of L-

functions. Recently, H. L. Montgomery [4] and M. N. Huxley [5] gave

two simplified proofs which are also based on the estimation of the

density of zeros of L-functions, and in their proofs, the approxi

mate functional equation of L-function and a mean value theorem on

the fourth moment of L-function are used.3' In this paper, a new

simplified analytic proof of the three primes theorem will be given

which is not based on the Vinogradov's estimation and the density

theorem of zeros of L-function, and only some well-known simple facts

on L-function are used.

(B) We use N to denote large integers, p, p-., p2, p, the ZTTI x

prime numbers, and e(x) = e . Let

Then

S(x, N) = I e(px) . (1) p<N

r(N) = I 1 = P1 + P2 + P3 = N

1 3 SJ(x, N) e(- Nx)dx (2) 0

denotes the number of representations of N as the sum of three

aRecently, K. Ramachandra [7] gave a simplified proof for the mean value theorem on the fourth moment of L-function.

primes. The three primes theorem is equivalent to r(N)>0 for N

is odd and sufficiently large. It is well known that the proof of

r(N)>0 is reduced to show that for any given positive integer c

and

we have

logcN < q < Nlog"cN , (h, q) = 1 , (3)

S (£ , N) « Nlog"3N . (4)

In this paper, we prove the following

Theorem. Let

T (x, N) = I A(n) logê(nx) . (5) 1 n<N n

If 1 < q< N and (h, q) = 1, then

T 1(H.H)«Nq-(1/ Z)log 1 0N + N^

4)q^4)W13/2)N . q (6)

We shall show that (4) can be drived by our Theorem, and

therefore the three primes theorem. We need the following well-

known lemma (Cf. [4], Theorem 6.2). Lemma 1. Let x(n) denote a character mod q. Then

nQ+k

I an X(n) n = nn+l

2 nQ+k

<(q + k) I |aj 2 , (7) n = nn+l

n

Q T . M - " Q

where I denotes a sum over all characters mode q. X

The proof of Theorem. For (h, q) = l, we have

V£,N)= I e(M) I A(n)log^ 1 °1 £=1 q n<N n

(«.,q)=l N = X.(modq)

+ I A(n) log J e(M) n < N H

(n,q)=l

T T - T I T ( X ) X(h) * , { N , X ) +o( log 2 N logq) , <t>(q) x '

where 4>(q) denotes the Euler function,

(8)

and

T(X) = I X(h) e(£) , h=l q

*i(N, X) = I A(n) x(n) log £ 1 n < N n

(9)

(10)

Since x(xQ)=y(q). where Xn denotes the principal character

mod q, |T(X)| < q~ if x X > and <(>(q) » q log" q , we have 0

T 1(J.N)«M9_« 1(N. X Q ) * 1 ^ I | V N , X ) | Sq X * X 0

+ log N log q (11)

It is easy to prove that for a>l»

ra + i°° i, 1 fa ' L' N b

VN- x ) ^ -r(s.x)V5

2-rri J L' N

- r (s> X) -5- ds , say (a) L s2

(12)

Let A(<N) be a constant which will be determined in the later.

Let

u(n) x(n) M(s, x) = I (13) n < A ns

where y(n) denotes the Mobius function. We use the identity [6]

- £ ( s , X) = -^'(s, x)(l-L(s, X) M(s, x))

- L ' ( s , X ) M(s, x) •

Set a = 1 + log_1N and B= [61og 2 N] . Then

- { M s , X) = ^ ( s , X) + f 2 ( s , x) + 0(N - 3)

(14)

(15)

where

^(s, X) = I A(n) X(n)

n < A n

f?(s, x) = I A(n) X(n)

A < n < 2BA n

Since for Re s = a, L(s, x) « l°gN and M(s, x) « logN, it follows from (14) and (15) that for Res = a,

- - (s, X) = ^{1 -LM) + f2(l -LM) - L'M + 0(N"2) .

From (12) and (17), we have

*i(N, X) =^-r f (fid-LM) + f 2(l-LM)-L'M)^ds

+ 0(N_1) .

If X=£XQ> the integrals corresponding to the first and third terms of the integrand may be shifted to the line Res = —. Therefore

1 *l(N« X) " o-r | W,

1 2TH J ( J ) 1

(f n ( l - L M ) - L'M) ^ - d s s^

2-rri J (a) f2(l -LM) — ds + 0(N _ I)

\ f (|f | + |f LM| + |L'M|) -^-r Ids Jin ' ' s s (1)

•|fj 11 - LM| - ^ |ds| +0(N _ 1: (a) L |s| 2

and by Holder inequality, we have

I I* (N, X ) | « N Jq J sup / I | f , | X * X 0 Res= iVx*X 0

+ N* sup ( I | f , | 4 ) Res=i \ x ^ X 0 /

i Ids! • sup ( I IM I 4 ) * } ( I | L | 2 )

Res=j\x^x0 / J(J)Vx^x0 /

+N*suP / I | H | 2 ) J J ( i ) ( I | L ' | 2 ) * ^ i Res=J V X * X 0 / H i } \ x * X 0 / | s | 2

+ N sup ( I \f2\2) sup / I | 1 - L M | 2 \ + qN_1

/ Re s=a \ x *Xn / Res=a \ x * X 0 X * X 0

(20)

Since

I |L(s . x ) | 2 « q|s| log2q|s | , (s = 1+ i t ) (21) X * X 0

I |L(s , X ) | 2 « q|s| log2q|s | , ( s = l + 1 t ) (22) X * X 0

which w i l l be proved in (D), we have

and

and

( I | L | 2 \ * i M « / J r i 0 g q

(i)\x*x0 / lsl2

( I | L M 2 ^ « ^ l o g 2 q (i)Vx*x0 / |s|2

(23)

(24)

Now we proceed to estimate the sums in the right hand side of (20)

by the use of Lemma 1. Hereafter we assume that q<N and A<N.

(a) By (16) and Lemma 1,

I 2

I Ifî+lt.x)! 2 5 (q + A) log3N X

(25)

(b) By (13) and Lemma 1,

I |M( j + i t , x ) |2 < (q + A) logN

X l (26)

(c) By (16),

ff(s.x)= Z , ^ an X(n)

n < A2 ns !a

nl < d(n) log'n ,

where d(n) denotes the divisor function, and therefore by Lemma 1

and I — ^ - « log x, we have n < x

I l^(l+it,x)|2 « (q + A2) log8N (27)

(d) By (13),

.2 •c r bn X(n)

n<A^ ns \ \ \ < d(n)

and therefore

I |M(1+ i t , X ) | 4 « (q + A2) log4N X l

(e) By (16), we have for Res = a, IB-1

I I V s ' X)T = 1 X X

y y Mn) x(n)

j = 0 2 j A < n < 2 j + 1 A "S

B-l )

j=0 2 JA<n<2 d + 1A n"

« (3. + log2N) log6N (29)

(f) For Re s = a, we have

c x(n) l 1-LM = I -=-= + 0(N"') , |c | < d(n) ,

A < n < 2BA nb n

2 3 and therefore by J d (n) « xlog x,

n < x

I |1 -LM|2 « (J+ log2N) log8N . (30 X A

From (20) and (23)-(30), we have

I |*1(N,x)l« ^ q 1 (q + A 2)hog 4N X*X 0

+ N(^ + log2N) log7N . (31

Take A=N* q*log^3/2^ N. Then it follows by (11) and (31) that

I|J, (N, xn) <K N> and therefore the theorem is proved.

From our theorem, it yields

Lemma 2. Suppose that c is an integer > 42. Then for

logCN < q < Nlog"CN , (h, q) = 1 (32

we have

^({J, N) « Nlog"4N . (33

(C) To prove (4), we shall need

Lemma 3 '. Let c be an integer > 46 and

T„(x, N) = I A(n) e(nx) . (34 u n<N

Then for logcN< q< N log~c N and (h, q) = l, we have

T0({|, N) « Nlog~2N . (35

This lemma was proposed by Prof. Din Xia Qi.

Proof. Let A = log N. Then

T}(±, IUAN) - ^ ( { j , N)

= l o g ( l + A ) T ( £ , N) + I A(n) log ^ ± M e(nx) . "0vq N < n < N+AN

(36) By Lemma 2 and (36),

log(l +A) T Q ( J , N) « Nlog"4N + AN log(l +A) . (37)

Since log( l + x ) > - x i f 0 < x < - , we have

T Q ( ^ , N) « A"1 N log" 4N + AN « Nlog" 2N (38)

This lemma is proved.

It follows by Lemma 3 and the summation by parts that for

l og c N<q<N log " c N , (h , q) = l , c > 42 , (39)

we have

I ^ e ( l £ ) « N l o g - 3 N . (40) 2<n<N l o 9 n Q

This is equivalent to (4), and thus the three primes theorem is

proved.

(D) The proofed of (21) and (22). Let x^X 0» H = [q|s|] and

F(x) = I x(n)- By Polya's theorem, we have H < n < x

F(x) « /q~logq

Therefore

f X(n) = f" dF(x) n = H+l ni + it JH xi + u

(l+it) F(x) 2 \(3/2) + it

dx

dx « |s| /cpogq - ^ « /|s| log q H x 3/2

(4 i ;

and by Lemma 1 , we have

I |L(}+it,x)|2« I ( I -*£jt * X n C X*Xn V n=1 n ^ l z

X*X0 X*X0

+ |sI log q

2 2 « (q + H)logH + q | s | log q « q|s| log q|s| (42)

(21) is proved. The proof of (22) is similar.

References

[1]. I. M. Vinogradov, Dokl. Akad. Nauk SSSR, 15 (1937) 291-294.

[2]. Ju. V. Linnik, Dokl. Akad. Nauk SSSR, 48 (1945) 3-7.

[3]. N. G. Tchudakov, Ann. of Math.; 48 (1947) 515-545.

[4]. H. L. Montgomery, Topics in Multiplicative Number Theory, Lee. Notes in Math.; Springer-Verlag, 227 (1971).

[5]. M. N. Huxley, The Distribution of Prime Numfeters, Oxford, Clarendon Press (1972).

[6]. Pan Cheng Dong and Din Xia Qi, Acta, Math. Sinica, 18, 4 (1975) 254-262.

[7]. K. Ramachandra, Ann. Scuola Norm. Sup. Pisa, CI. Sci.; 4 (1974) 81-97.

(See Acta Math. Sinica, 20 (1977) 206-211).


Let

where

An Elementary Method in Prime Number Theory

R. C. VAUGHAN

Mathematics Department, Imperial College, Cromwell Road, London SW7 2AZ, England

1. INTRODUCTION

T(Y,Q)= I - f - r m a x | « / K X ) A r ) | (1)

4>{X,X)= I Mn)X(n) (2)

and X* denotes summation over primitive characters modulo q. An estimate for T is an essential ingredient in the Bombieri-Vinogradov theorem on primes in arithmetical progressions. Also, let

H,(Y,Q)= I - f - I * max \Mr{X,x)\ (3) q*Q<P(q) X X*Y

where

Mr(X,X)= I n(n)X(n). (4) ( n . r ) = l

The purpose of this note is to adumbrate proofs of the following two theorems based on ideas contained in [5], [6].

THEOREM 1. Let Q s* 1, Y s= 2, if = log YQ. Then

T(Y,Q)«(Y+Y*Q+YiQ2)£4. (5)

342 R. C. VAUGHAN

THEOREM 2. Let Q 3= 1, Y 3= 2, r 3= 1, i? = log YQ. 77te«

H r ( y , 0 ) « ( y + d ( r ) y | o + y-io2)if4. (6)

Theorem 1 combined with the Siegel-Walfisz theorem easily gives

THEOREM 3 (Bombieri-Vinogradov). LefQ 3=1, Y**2,& = \og YQ. Then

I sup U(X,q,a)-^~\«AY{\ogY)~A+Y{Q<g*. (7)

( a , 4 ) - l . X « Y

Similarly Theorem 2 gives

THEOREM 4. Let Q 3= 1, Y 3= 2, if = log YQ. 77ien

I sup Z M (n ) « A Y(log Y)~A + Yl Qi?4. (8) X ^ V n«a(mod<i)

2. PROOFS OF THEOREMS 1 AND 2

LEMMA 1. Suppose that am (m = l , . . . , M ) and fc„(n = 1 , . . . ,7V) are complex numbers. Then

M N

q*.o4>(q)

a v* I I ambnX(mn)\«((M + 02)(N + Q2)l\am\2l\bn\2)\

This is an immediate consequence of the large sieve inequality (see, for example, Gallagher [1], or (1.4) of [2]) and Cauchy's inequality.

LEMMA 2. On the premises of Lemma 1,

a M N

I —rrsup q*Q<p(q) x X*Y

I I ambnx(mn)\ t - i n - i i

1

«((M + 02)(N + 02)l\am\2l\bn\2Y log YMN. (9)

Proof. Let

J-ao a

and y>0 . Define £(£)» 1 when 0«/3 < y and 5 ( 0 ) - 0 when 0 >y. Then C > 0 and it is easily seen that f or A Ss 1, 0 3= 0, 0 =* y, one has

5(^)=f A e ; e - -^da + 0(A-1 |y-^r).

23. AN ELEMENTARY METHOD IN PRIME NUMBER THEORY 3 4 3

Let y = log ([X] + 5), /3 = log mn. Thus A

I I ambnx(mn)=\ Y.Zamm'abnn'a'}((mn)—-—da m n •*—A m n \^0C

mn ^X

+ o(x4- ,H|am6n | ) . \ m n 1

The desired conclusion now follows easily from Lemma 1 on taking A = YMN.

If Q2> Y, then Theorem 1 follows at once from Lemma 2 on taking M= 1, a\ = 1, b„ = A(n). Hence it may be assumed that Q =s y.

Let

u = min(Q2 , y*. yCT2). (10)

By applying Lemma 2 as in the case Q2> Y it is easily seen that

I T T T I * sup |<M**) |«(« 2 (? + MQ2)if2. (11)

Consider the identity

I A(«)/(n) = S 1 - S 2 - S 3 (12) u<n*X

where

5 j = I I /x(m)(log n)f(mn), (13)

S2= I I c„/(m«), c m = I M(fl)A(i), (14) mû2 n^X/m a*u,b*u

ab = m

S3= I I rmA(n)f(mn), rm = I /*(«/), (15) m>u n>u d\m

which is most readily obtained by inspecting the coefficients in the Dirichlet series identity

(-j(s)-F(s)) = G(s)(-C'(s))-F(s)G{s)tts)

-{((s)G(s)-l)(-£(s)-F(s)) (16)

where

F(s)= I A(n)n~', G(s) = I n^n". (17)

344 R. C. VAUGHAN

On writing/(«) = ^(n) in (12), one sees that it suffices to show that for/ = 1, 2, 3 the sum

r , = I - f r L * sup \S,\

satisfies (5) with T replaced by T,. Note that the terms in £ „ < x A(")*(«) with n «s u can be taken care of by (11).

By (15),

T3 I T3(M)

where M = {2ku : it - 0 , 1 , . . . ; 2 V « Y) and

r 3 ( M ) = I 7 7 - r I * sup \S3(M)\

with

5 3 (M)= I I TMA(/I)*(/TOI). M < m « 2 M u < n « X / m

By Lemma 2,

r 3 ( M ) « ( ( M + 0 2 ) ( y M _ 1 + 0 2 ) I d(mf I A(n)2Y log Y \ m*2M n « V / M /

l 1 i i - . 2 w . _ - v \ 3 « ( y + y5M5o + rM~5o + y5oz)(iog y)

which easily gives the desired conclusion. By (13), on writing log n = | " (da/a) and interchanging the order of

summation and integration, one obtains

J, x i

Z ti(m)xim) I * ( n ) — . 1 m<min(u.X/a) o«n«:X/m <*

Using the Polya-Vinogradov inequality (Schur's proof [3] is elementary) when q > 1 gives one

r1«(y+«o l)( iogy)2

which with (10) again gives a suitable estimate. The expression T2 is estimated by combining the above arguments. The

sum S2 is divided into two parts

S2 = S2 + S2

where S2 contains the terms with mû and S'i the terms with u < m =s u2. Then Si is treated like 5i and S2 like S3. This provides an appropriate upper bound for T2 and completes the proof of Theorem 1.

23. AN ELEMENTARY METHOD IN PRIME NUMBER THEORY 345

Theorem 3 follows from Theorem 1 in the same manner that Corollary 1.1.1 of [4] is deduced from Theorem 1 therein.

2. PROOFS OF THEOREMS 2 AND 4

The proof of Theorem 2 is similar to that of Theorem 1, using instead the identity

I M(n)/(n) = 2 5 1 - 5 2 - 5 3 (18) n«=X

where

5,= I n («)/(«) (19)

52= Z 2 I cmf{mn), cm= Z fi(a)n(b), (20) ab = m

53= I Z v ( n ) / ( m « ) , r m = Z /*(<*)• (21) m>u n>u </|m

This is an immediate consequence of the identity

- i - = 2G(s) - G(s)2£(s) - U(s)G(s) - l)(-±--G(s)) (22)

where G(s) satisfies (17). The case Q2 > Y of Theorem 2 can be treated as in the proof of Theorem

1. Let u satisfy (10). Then, as before, (11) holds with t(/(X, x) replaced by Mr(X,x). Now in (18) let f(n) = x(n) when (n, r) = l and f(n) = 0 when (n, r) > 1. Then it suffices to bound

T,= Z T £ T Z * sup |S,| (7 = 1,2,3).

The sum T\ can be bounded at once by appealing to the analogue of (11). Also the sum T3 can be estimated in the same manner as the corresponding sum occurring in the proof of Theorem 2. Similarly T2 can be estimated by dividing 52 into two parts 52 and 52 according as m « u or m > u. Thus r 2 «£ 7*2 + T'{ where T'2 and T'i correspond to T2 with 52 replaced by 52

and Si' respectively. Now 7"i' can be treated in the same manner as T3. It remains, therefore, to consider T'2-

86

346 R. C. V A U G H A N

By the Polya-Vinogradov inequality, when * is a non-principal character with modulus q > 1,

I X(n) = l»(d)x(d) I xim) n*iZ d\r m*Z/d

( n . r ) - l

Hence «-d(r)q* log q.

T'2«(Y + d(r)uOh^2

The proof of Theorem 2 is completed by observing that, by (10), wQ2=s

The proof of Theorem 4 is rather more involved than that of Theorem 3, there being an extra difficulty in the reduction to primitive characters.

Observe that

sup sup £ n(n) a X « y l n « X

n • a(mod q)

Z sup sup X fi{m) \.

(fr,q/r)=l m»6(modq/r) (m,r)=l

Hence

where

£ sup sup q < 0 a X * V

I nM * I F,(Y/r,Q/r)

n"o(mod q)

Fr(Y,Q)= I sup sup X /X(H) q « 0 a X * V n « X

(a,q)»l n«a(modq) <n,r)-l

When(a ,q) = l ,

1 I M(«) = T7~: I *(«) I x(n)/Lt(w).

n * X Ûj)*,™*!. , n * X nâdnodq) (n,r)»l

(n,r)- l

Therefore

whence

where

I /*(n) n « X

nHa(mod q) (n.r)-l

* I I* I ^-(/I)M(M) n « X

(n,rq/d)-l

F , ( y , 0 ) « I -—GAYtQI'k)

G,{Y,Q)= I T T - r r s u p l I X ( H ) M ( " ) | q*.Q<p\q) x x « y l n « x I

(n.r)-l

(23)

(24)

(25)

23. AN ELEMENTARY METHOD IN PRIME NUMBER THEORY 3 4 7

Let R^Q. Then, by partial integration, (3) and (4),

Gr(Y,Q)-Gr(Y,R)*sQ-lHr(Y,Q)+\ a~2Hr(Y,a)da.

Therefore, by Theorem 2,

Gr(Y,Q)-GAY,R)«(YR-1 + Y*Q)2A + d(r)Ys&5. (26)

Suppose that q «s (log Z ) A and * is a character modulo q. Then a standard application of the theory of Dirichlet L-functions gives

I x(n)p(n)«Ad(r)Zexp(-c(logZf)

(# i . r ) - l

where C is a positive constant. Hence, by (25),

Gr(Y, (log Y)B)«Bd(r)Y exp( -k ( log Yf).

This combined with (26) and a suitable choice of B gives

Gr(Y, Q) «A (KSTA _ 4 + Y*QZ*) d(r).

Hence, by (24),

Fr(Y, Q) «A (Y£-A-2+ yÔif4) d(r).

Therefore, by (23), when Q =s Y\

«AY#~A+Y*Q&*. I sup q « 0 a,X

Z M(«) F I < X

X < V n—a(mod(j)

The proof of Theorem 4 is completed by noting that the conclusion is trivial when O > Y*.

REFERENCES

[1] Gallagher, P. X. The large sieve. Mathematika 14 (1967),.14-20.

[2] Montgomery, H. L. and Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119-134.

[3] Schur, I. Einige Bemerkungen zu der vorstehenden Arbeit des Herr G. Polya: Uber die Verteilung der quadratischen Reste und Nichtreste, Gottinger Nachrichten 1918, 30-36.

[4] Vaughan, R. C. Mean value theorems in prime number theory. /. London Math. Soc. (2) 10 (1975), 153-162.

88

348 R. C. VAUGHAN

[5] Vaughan, R. C. Sommes trigonometriques sur les nombres premiers. Comptes Rendus Acad. Sci. Paris, Serie A, 285 (1977), 981-983.

[6] Vaughan, R. C. On the distribution of ap modulo 1. Mathematika 24 (1977), 135-141.

(See "Recent Progress in Analytic Number Theory", edited by H. Halberstam and C. Hooley, Acad. Press, 1981, pp. 241-248.)

Noted by Pan Cheng Biao: From (12), we can easily derive the

following estimation for S(a) = \ A(n)e(na), namely if |a - -| < q , n< x q

i/o //*% i/o 1/0 7/0

(a,q) = 1, then S(a) « (xq +x +x q )log x. We may assume

without loss of generality that q<x. Since

. x 1 I max m< y to

I e(mna) « \ m i n ( - , ) « (xq + y + q ) l o g q y , u < n < x / m m < y m ||ma||

where | |£ | | denotes the l e a s t d i s t a n c e from E, to an i n t e g e r , we have

from (13 ) , (14) and (15) wi th f (n ) = e (na) the fo l lowing e s t i m a t i o n s

S « l o g x I m< u a)

J e(mna) to< n < x/m

, -1 ^ 2 « (xq + u + q ) l o g x ,

S 2 « log x I 2 I e (mna) m < u n < x/m

-1 2 2 « (xq + u + q ) l o g x

and

S, « log x max u < M < x / u

J T J A(n)e(mna) M<m<2M m u < n < x/m

, 5/2 „ l / 2 « log x max M u < M < x / u \ M < m < 2 M

I £ A(n)e(mna) u < n < x/m

1/2, 3 « x log x max

u<M< x/u u< n . < x/M I u < n 2 < x/M

1 e ( ( n - n„)ma M< m < 2M m < x / n . m< x /n„

90

log x max max I £ min(M, u<M< x / u u< n < x/M û < n 2 < x/M || ( n ] - n 2 ) a |

« x l / Z l o g J x max ( M + £ min( *- , - ) u < M< x /u \ 1 < m< x/M '" | |ma"

, - 1 / 2 - 1 / 2 1/2 1/2 , , 7/2 « (xq ' +xu + x q ) l o g x .

2/5 Take u = x . The assertion follows.

ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 99-104 (September 17, 1997) S 1079-6762(97)00031-0

A COMPLETE V I N O G R A D O V 3-PRIMES THEOREM U N D E R THE R I E M A N N HYPOTHESIS

J.-M. DESHOUILLERS, G. EFFINGER, H. T E RIELE, AND D. ZINOVIEV

(Communicated by Hugh Montgomery)

A B S T R A C T . We outline a proof tha t if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all bu t a finite number of cases, an intermediate lemma, and an extensive computation.

1. INTRODUCTION

By "The 3-Primes Problem," we mean: can every odd number greater than 5 be written as a sum of three prime numbers? This problem was first successfully attacked by Hardy and Littlewood in their seminal 1923 paper [6]; using their Circle Method and assuming a "Weak Generalized Riemann Hypothesis," they proved that every sufficiently large odd number could be so written. The second author has calculated [4] directly from that paper that "sufficiently large," assuming the "full" Generalized Riemann Hypothesis (GRH below, i.e., that all non-trivial zeros of all Dirichlet L-functions have real part equal to 1/2), is approximately 1050. In 1926 Lucke [11], in an unpublished doctoral thesis under Edmund Landau, had already shown that with some refinements the figure could be taken as 1032.

In 1937 Vinogradov [15] used his ingenious methods for estimating exponential sums to establish the desired asymptotic result while avoiding the GRH entirely. However, the numerical implications of avoiding the GRH are substantial: in 1956 Borodzkin [1] showed that sufficiently large in Vinogradov's proof meant numbers greater than 3 3 « 107000000 T h i s figure h a s

since been improved significantly, most recently by Chen and Wang [2], who have established a bound of 1043000, but in any case this figure is far beyond hope of "checking the remaining cases by computer."

If, however, we return to the original stance of Hardy and Littlewood by assuming the truth of the GRH while at the same time using some of the refined techniques of primarily Vinogradov and Linnik [10], and using an extensive computer search, we do indeed arrive at the following: T h e o r e m . Assuming the GRH, every odd number greater than 5 can be expressed as a sum of three prime numbers.

Received by the editors February 26, 1997. 1991 Mathematics Subject Classification. Pr imary 11P32. Key words and phrases. Goldbach, Vinogradov, 3-primes problem, Riemann hypothesis.

©1997 American Mathematical Society

100 J.-M. DESHOUILLERS, G. EFFINGER, H. TE RIELE, AND D. ZINOVIEV

The proof of this result falls naturally into three parts: an asymptotic theorem handling all but a finite number of cases, a lemma assuring the existence of primes relatively near unchecked odd numbers, and a computer search for 2-primes representations of the remaining differences. We now outline each of these parts.

2. T H E ASYMPTOTIC THEOREM

T h e o r e m (Zinoviev). Assuming the GRH, every odd number greater than 1020 is a sum of three prime numbers.

We discuss here briefly the main ideas behind this result; for complete details see [16].

Fix N > 9. We are interested in the number of triples (pi,P2,P3) of prime numbers which satisfy the equation

(1) N=Pl+P2+P3-

Following [10] we introduce the function

J(N)= X I l°g(Pl)l0g(P2)log(p3), Pl+P2+P3=N

where the sum ranges over all triples of primes (> 2). If J{N) > 0 then there is at least one solution of (1). Here by A(n) (n is always a natural number) we denote von Mangoldt's function: A(n) = log(p) if n = pk (p is prime), and A(n) = 0 otherwise. For any real number a set

S(a) = J2Mn)e-27riane-n/N. n > i

Then we have

5(a) = Y^log{p)e-27Tiape-p/N +eN°-5\og2{N), P > I

where \9\ < 1. Clearly, for any integer m

f Jo

\^da: ) X i f m = ° ' 0 if m + 0.

Changing the order of summation and integration (see [10]), for some new real 6 (|0| < 1) we obtain

/

l — W

S3(a)e27riaNda + ON1-6 log3{N), -w

where w = w(N) is a small number defined later. We will express J(N) as a sum of the leading term and the remainder. Estimating the remainder from above, we will show that it is less than the leading term when N > 1020. We then conclude that J(N) > 0.

Following Linnik and Vinogradov, we subdivide the interval [—w, 1—w] into the disjoint union of subsets E[, E", E2. Our main idea is to refine this subdivision. In particular we change the set of "major arcs", which in our case is E[, making the intervals from this set smaller. We do it as follows.

L e t Q = [l.llog2(JV)], r = 4900 log4(JV), w = 1/T.

A COMPLETE VINOGRADOV 3-PRIMES THEOREM 101

Denote by E(a, q) (where if q > 1, then (o, q) = 1, 0 < a < q, and if q = 1, then a = 0) the interval

— £ _1 qr' q qr

Then

[—w, 1 — «;] u u 0 < q < r 0<a<ij L

(?,<*) = !

_L_ o 1 - H q qr

Let £1 = {E(a, q), q < Q} and E2 = [-w, l-w\-Ex. Finally, denote by E{ the set of intervals E\ with smaller length

a 21og(AQ a 21og(JV) q <f>(q)N ' q + ^q)N

and set E'{ = Ex - E[. We split the integral J{N) into two integrals: over E{ (the leading term) and

E" U E2 (the remainder). The following lemma is used to estimate the remainder term.

Lemma. For any a € E" U E2, and for any N > 1020 (not necessarily odd), GRH implies that

|5 (Q)l<°-18Ic^V? The proof of this lemma uses the Riemann-Hadamard method which involves

summation over the zeroes of L-functions. The leading term is treated using the circle method of Hardy and Littlewood, as

used by Vinogradov and Linnik.

3. A N INTERMEDIATE LEMMA

Now, by the asymptotic theorem, our problem is reduced to considering odd numbers which are < 1020. For these, we need the following:

Lemma. / / the GRH holds and if 6 < n < 1020, then there exists a prime number p such that 4 < n-p< 1.615 x 1012.

Proof. The conclusion of the lemma obviously holds for n < 1012, say. For larger n, we apply Schoenfeld [13], equation (6.1). Let 6(n) = Ylp<n^°&P' if the GRH holds, and if n > 23 x 108, we have

|0(n) — n\< — v^n(logn - 2)logn.

Just suppose that there is no prime in the interval (n — h, n] except possibly for two of the six consecutive numbers from n — 5 through n; then we have

21ogn > 9(n) - Q(n - h) = (9(n) - n) - (0(n - h) - (n - h)) + h,

whence by the above,

h < — v/n(log n — 2) log n + 2 log n.

Since n < 1020, we get h < 1.615 x 1012. We conclude then that there must be a prime p such that 4 < n — p < 1.615 x 1012. •


We note here that the GRH actually implies an estimate on |G(n) — n\ which has a single log factor; see Ivic [7] for example. However, the second author, in working through the details of such an estimate, found that the constant obtained was large enough so that, at the level n = 1020, Schoenfeld's estimate gives a slightly better numerical result.

4. T H E COMPUTER SEARCH FOR 2-PRIMES REPRESENTATIONS

Finally, then, if n is an odd number < 1020 and p is as in the previous lemma, then m = n — p is even and < 1.615 x 1012. But for m we have the following:

T h e o r e m (Deshouillers and te Riele). Every even number 4 < m < 1013 is a sum of two prime numbers.

For a complete exposition of this and related results, see [3]. Let pi be the ith odd prime number. The usual approach [5], [14] to verify the Goldbach conjecture on a given interval

[a, 6] is to find, for every even e € [a, 6], the smallest odd prime pi such that e — pi is a prime. An efficient way to do this is to generate the set of primes

Q(a, b) — {q | q prime and a — ea < q < b},

where ea is chosen in a suitable way, and to generate the sets of even numbers £o C £\ C £2 C • • •, defined by So = 0,

£i+1 = £ j U (Q(a, b) + p i + i ) , i = 0 , 1 , . . . , 1

until £j covers all the even numbers in the interval [a, b] for some j . The set Q(a, b) is generated with the sieve of Eratosthenes: this is the most time-consuming part of the computation. For the choice of ea it is sufficient that ea exceeds the largest odd prime used in the generation of the sets £j. In the computations checking the Goldbach conjecture up to 4 x 1 0 u [14], the largest small odd prime needed was p 4 4 6 = 3163 (this is the smallest prime p for which 244885595672 — p is prime).

A more efficient idea, which we have implemented, is to find, for every even e G [o, b], a prime q, close to a, for which e — q is a prime. To do that efficiently, a set of k consecutive primes <?i, 92, • • • ><2fc close to a is generated, for suitably chosen k, and a large set V of all the odd primes up to about b — a is precomputed (with the sieve of Eratosthenes) in order to check the numbers e — q for primality. For the actual check of the interval [a,b], we generate the sets of even numbers f o C f i C f 2 C . . . , defined by JF0 = 0,

Ti+l = Ti U {V + qi+1), i = 0 , l , . . . ,

until Tj covers all the even numbers in the interval [a, 6] for some j . In our experiments, we have chosen the intervals [o, 6] to have a fixed length of 108. The largest possible prime we may need in the set V lies close to b — qi. By the prime number theorem, 91 « a — k log a, so that b — qi « 108 + k log a. For the choice of k we notice that the density of the odd primes among the odd numbers up to 108 is about 0.115 (there are 5761454 odd primes below 10s). This means that a proportion of about 0.885 of the even numbers in [o, b] is not covered by the set T\ = V + q±\ if the primes up to 108 were uniformly distributed, which they are not, a proportion of about 0.8852 of the even numbers would not be covered by Ti- After 151 steps, this proportion is reduced to below 10 - 8 . It turned out that k = 360 was sufficient

x By Q(a, b) + P i + i we mean the set: {q + Pi+\\q 6 Q{a, &)}.

A COMPLETE VINOGRADOV 3-PRIMES THEOREM 103

for our experiments. For a « 1013 this implies that the largest prime in the set V must have a size close to 108 + 104.

In the first approach, a small set of small primes up to 5000, say, has to be available, and for each interval [a, b] to be treated, all the primes in [a, b] have to be generated. In the second approach, a large set of small primes up to about 108 + 104

has to be generated (only once), and for each interval [a, b] to be treated, one has to find the largest k primes < a. Of course, this is much cheaper than to find all the primes in the interval [a, b\. The price to pay is that for each e G [a, b] some prime p is found for which e — p is prime, but in general this p is neither the smallest nor the largest such prime.

For the actual generation of k primes close to a we have used Jaeschke's computational results [8], stating that if a positive integer n < 2152302898747 is a strong pseudoprime with respect to the first five primes 2,3,5,7,11, then n is prime; corresponding bounds for the first six and seven primes are 3474749660383 and 341550071728321, respectively.

We have implemented the second approach on a Cray C98 vector computer and verified the Goldbach conjecture for all even numbers > 4 x 1011 and < 1013. After the generation of k primes near a,, the actual verification was carried out by sieving with a long array of 64-bit integers called ODD, where each bit represents an odd number < 108 + 104, the bit being 1 if the corresponding odd number is prime, and 0 if it is composite. Generating Ti+i from Ti amounts to doing an "or" operation between one long array of integers and a shifted version of the array ODD. This can be carried out very efficiently on the Cray C98. In one typical run, we handled 5000 consecutive intervals of length 108. Close to 1013 the time to generate 5000 x 360 large primes was about 2600 CPU-seconds, and the total sieving time was about 5040 seconds. The total time used to cover the interval [4x 1011,1013] was approximately 53 (low priority) CPU-hours. The largest number of large primes which we needed was 328: for e = 7379095622422 and first prime qx = 7378999992031 it turned out that e - qt is composite for i = 1,. .. ,327, and prime for i = 328 (g328 = 7379000002739 and e - g328 = 95619683).

ACKNOWLEDGMENT

The second author wishes to thank Paul T. Bateman and Marshall Ash for helpful correspondences on this topic.

REFERENCES

1. K. G. Borodzkin, On I. M. Vinogradov's constant, Proc. 3rd Ail-Union Math . Conf., vol. 1, Izdat. Akad. Nauk SSSR, Moscow, 1956. (Russian) MR 20:6973a

2. Jingrun Chen and Tianze Wang, On the odd Goldbach problem, Acta Math. Sinica 32 (1989), 702-718. MR 91e:11108

3. Jean-Marc Deshouillers, Herman te Riele, and Yannick Saouter, New experimental results concerning the Goldbach conjecture, to appear.

4. Gove Effinger, Some numerical implication of the Hardy and Littlewood analysis of the 3-primes problem, submitted for publication.

5. A. Granville, J. van de Lune, and H. te Riele, Checking the Goldbach conjecture on a vector computer, Number Theory and Applications (R.A. Mollin, ed.), Kluwer Academic Publishers, 1989,423-433. M R 93c: 11085

6. G. H. Hardy and L. E. Littlewood, Some problems of 'Partitio Numerorum}. Ill: On the expression of a number as a sum of primes, Acta Mathematica 4 4 (1922), 1—70.

7. A. Ivic, The Riemann zeta-function, J. Wiley and Sons, 1985. M R 87d:11062

96


8. Gerhard Jaeschke, On strong pseudoprivn.es to several bases, Math. Comp. 6 1 (1993), 915—926. M R 9 4 d : 1 1 0 0 4

9. A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, 1993. M R 94a:11001 10. U. V. Linnik, The new proof of Goldbach-Vinogradov's theorem, Mat . Sb. 19 (1946), 3-8. M R

8:317c 11. Bruno Lucke, Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems, Doctoral

Dissertation, Gottingen, 1926. 12. Paulo Ribenboim, The book of prime number records, Springer-Verlag, 1988. MR 89e:11052 13. Lowell Schoenfeld, Sharper bounds for the Chebyshev functions @(x) and ^(x), Mathematics

of Computat ion 3 0 (1976), 337-360. M R 56:15581b 14. Matt i K. Sinisalo, Checking the Goldbach conjecture up to 4 X 10 1 1 , Mathematics of Compu

tat ion 6 1 (1993), 931-934. M R 94a:11157 15. I. M. Vinogradov, Representation of an odd number as a sum of three primes, Comptes

Rendues (Doklady) de l 'Academy des Sciences de l 'USSR 15 (1937), 191-294. 16. Dmitri i Zinoviev, On Vinogradov's constant in Goldbach's ternary problem, Journal of Num

ber Theory, 6 5 (1997), 334-358.

MATHEMATIQUES STOCHASTIQUES, U M R 9936 CNRS-U.BORDEAUX 1, U . V I C T O R SEGALEN

BORDEAUX 2, F33076 BORDEAUX C E D E X , F R A N C E

E-mail address: dezou®u-bordeaux2. f r

DEPARTMENT OF MATHEMATICS AND C O M P U T E R SCIENCE, SKIDMORE C O L L E G E , SARATOGA

SPRINGS, NY 12866

E-mail address: e f f ingerQskidmore.edu

C E N T R E FOR MATHEMATICS AND C O M P U T E R SCIENCE, P . O . Box 4079, 1009 AB AMSTERDAM,

T H E NETHERLANDS

E-mail address: h e r m a n . t e . r i e l e a c v i . n l

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E-mail address: zinovidQmemotec.com

II. REPRESENTATION OF AN EVEN NUMBER AS

THE SUM OF TWO ALMOST PRIMES

(ELEMENTARY APPROACH)


The Sieve of Eratosthenes and the Theorem of Goldbach

Viggo Brun

§1. The theorem of Goldbach is well-known that one can write

every even number as a sum of two prime numbers. In a letter of

1742, Euler has written: "I believe it is a completely acceptable

theorem, although I cannot prove it." This theorem has still not

been proved, and it is the same about the following theorem: The

sequence of the twin prime numbers') is infinite. In an address

delivered at the International Congress of Mathematics, Cambridge,

1912, E. Landau had said that he regarded these problems as

"unattainable problems in modern science."

However, one has now a starting point for the treatment of

these problems, after which one has discovered that the prime

numbers of Goldbach and twin prime numbers can be determined by a

method analogous to that of Eratosthenes. The first who had paid

attention to this fact should be Jean Merlin.2)

The method consists of a double employing the Eratosthenes

sieve. Let us, for example, give the partition of the even number

26. We write the following two sequences of numbers

0.1 2 3 4_ 5 6. 7 8. 9 10 11 1_2 13 14 1_5 ]6^

17 ^ 8 1 9 20 jH 22 23 2 4 2 5 26

26 25 24 23 22 21 20 19 1J5 17 1_6 1_5 14 13

12 11 10 9 8 7 6 5 4 3 2 1 0 .

l)That is to say that the couples of the prime numbers having the difference 2. See P. Stackel in "Sitzungsberichte der Heidelberger Akademie Abt. A., Jahrg; 1916, 10 Abh.

2)see Bulletin des Sciences mathematiques T. 39, I partie, 1915. See also Viggo Brun in "Archiv for Mathematik og Naturvidenskab" 1915, B. 34, nr. 8: "Uber das Goldbachsche Gesetz und die Anzahl der Primzahlpaare."

The prime numbers not exceeding /26 are 2, 3 and 5. We efface

the numbers of the form 2A, 3A and 5A in our two sequences. The

sum of a number of the first line and the number immediately below

in the second line is 26. If these two numbers are not effaced,

they are prime numbers, and give then a Goldbachian partition of 26.

It is not necessary to write the second sequence. One can only

choose the numbers 26 and 0 of the first sequence as the starting

points of the effacements. By this method we obtain all the parti

tions of an even number x into a sum of two prime numbers lying

between /x" and x -/<". On choosing 0 and 2 as the starting

points, we can determine the twin prime numbers. We do not know

if a treatment by this method can lead to a proof of these theorems;

but we see that the method can at least lead to very profound

results.

§2. We study at first the method of Eratosthenes, on giving it

the following form:

Suppose that the series:

0 1 2 3 4 5 6 7 8 9 1 0 . . . x

0 2 4 6 8 10 . . .

0 3 6 9

0 pn 2pn 3pp . . . Apn

are given-, where x denotes an integer and pn the n-th prime

number:

Pn * ^ < pn+l '

and A an integer:

Apn < x < (X+l)pn -

The terms of the first series, which are different from all the

terms of the other series, are the prime numbers lying between /x"

and x and the number 1.

These are the terms not effaced by the Eratosthenes sieve. We

generalize, on studying the following arithmetical progression

A A + D A+2D ...

a-| a1+p1 a]+2p] . . .

a a + D a + 2 D . . . r r Kr r Hr

The progressions are extended from 0 to x. D denotes an

integer prime to the prime numbers p, ,...,p (successive or not,

but different).

A and a, a are integers:

0 < A < D , 0 < a. < p. .

We raise the following problem:

How many terms different from all the terms of the other lines

does the first line contain?

We denote this number by

N(A, D, x, a r p1 ,ar> pr)

or often more briefly by

N(D, x, p r...,p r) .

We obtain the fundamental formula:

N(A, D, x, a r p1,...,ar, pr)

= N(A, D, x, a.j, P1, • • • .ar_-,. Pr_-|)

- N(A', Dpr, x, a r p1,...,ar_1, p r - 1)

where

0 < A'< Dp

or more b r i e f l y

N(D, x, p r . . . , p r ) = N(D, x, p ] , . . . , p r _ 1 )

- N(Dpr, x, p 1 , . . . , p r _ ] ) (1)

on studying at first our arithmetical progressions up to the pro

gression a -J + Ap i, and on subjoining then the progression

a +Ap . Suppose that N(A, D, x, a,,p, ,...,a _, ,p _,) is known.

We deduce N(A, D, x, a,,p,,...,a ,p ) from it on subtracting the

number of the terms of the last progression, which are identical to

the terms of the first progression, but not identical to the terms

of the intermediate progressions.

We see that the number is equal to N(A', Dp , x, a,, p,,...,p _,)

on noting that the terms of the last progression a + Ap , which

are identical to the first progression A + uD, are the terms

between 0 and x of the arithmetical progression

A' A' +Dp r A' +2Dpr . . . ,

where

0 < A' < Dpr ,

A' being the smallest positive term of the progression.

The indeterminate equation

ar + Apr = A + pD

or

p A - D u = A - a yr r

always has, as one knows, solutions, because p and D are

relatively prime. The solutions are

A = AQ + tD , v = VQ + tpr ,

whenever AQ, yQ are solutions and t runs through the values 0,

±1,±2

The terms of the last progression, which are identical to the

terms of the first progression, are then all the terms

ar+ A P r = ar

+ Vr + t D pr ' where t = 0' ±1, ±2>---

These are the terms of an arithmetical progression having the

difference Dp .

We define particularly N(A, D, x) or briefly N(D, x) as the

numbers of the terms between 0 and x of the progression

A A+D A + 2D . . . A+AD ,

where

0 < A < D , A + A D < x < A + (A+l)D .

Hence we deduce that

A + 1 = N(D, x) = £ + 6 , where -1 < 6 < 1 .

We give an example, choosing

A = 2 D=7 x = 60 8 ^ 2 p1 = 2 a2 = l p 2=3 a, = 4 p 3=5

(A) 2 9 16 23 30 37 44 51 58

(B) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

36 38 40 42 44 46 48 50 52 54 56 58 60

(C) 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

52 55 58

(D) 4 9 14 19 24 29 34 39 44 49 54 59

The numbers of (A) which are different from the numbers of (B)

and (C) are 9, 23, 51. We subjoin then the progression (D). The

numbers of (A) and (D), which are identical, are 9 and 44, having

the difference 7«5. We obtain then

N(7, 60, 2, 3, 5) = N(7, 60, 2, 3) - N(7'5, 60, 2, 3)

or 2 = 3-1.

From the formula (1) we deduce the fol lowing

N(D, x , P ] p r) = N(D, x) - N(Dp r x) - N(Dp2, x, P ] )

- . . . - N(Dpr, x , p1 p r - 1 )

(2)

and

N(D, x, pr...,Pr) = N(D, x) - N(Dpr x) - ... - N(Dpr> x)

+ N(Dp2pr x)

+ N(Dp3pr x) + N(Dp3p2, x, p7)

+ . . .

+ N(Dprp15 x) + N(Dprp2, x, p ^

+ ... + N(Dprpr_r x, pr...,pr_2)

(3)

We give the last formula a concise form

N(D, x, p„...,pj = N(D, x) - I N(Dp , x) a< r

+ 1 I N(DPaPb> x« Pr-'-'Ph-i)-

a< r b< a

(3')

When the question is to determine a lower bound for N(D, x,

p, ,...,p ) we can set aside as many positive terms as we want in

the formula (3). One can choose these terms in several different

ways3), for example, the terms which lie on the right of a vertical

line. In general we obtain the formula

N(D, x, p,,...,pj > N(D, x) - I N(Dp , x) a < r

+ 1 1 N(DPaPb' x' Pv-'Pb-l) ' (4)

ul

where we have chosen for pp. a domain a), which lies in the

interior of the following domain

P2PT

P3PT P 3P 2

prpl prp2 ••• prpr-l

3) See: "Nyt tidsskrift" 1918: Une formule exacte pour la determination du nombre des nombres premiers audessous de x, etc. by Viggo Brun.

On applying the formula (4) twice we obtain the new formula

N(D, x, pr...,pr) > N(D, x) - I N(Dpa, x) a < r

+ I I ( N ( °P a Pb ' x> " I N(DP aPbPc' x> ) c , and w„ denotes the domain for p p..

On continuing and applying

N(d, x) = 4 + 6 , where -1 < 9 < 1 ,

we obtain at last the general formula

£ N ( D , x, Pl Pr> > i - i f + n^Vh(i- i J-) a < r Ka u>, HaKb v c < b Hc /

H H — 1 — (i - y .1) L,<L L.,L P^PkP.Pn V A A PQ / u ; u 2 Pa^bPc^d e < d Ke

RD x (5)

where R denotes the number of terms, and where o^ < OK etc.

We can also give the formula (5) the fol lowing form, on

supposing par t i cu la r l y p-,=2, Pp = 3, P3 = 5 e t c . :

N(D, x, 2, 3 , 5 , . . . , p r ) > J 1 1 1 2 " 3 " 5

+ . . .

p -2 p .3 2 p .5

-1.1 ' 1 ' 3

1 3-2

Pr-7

1 1 1 ~ 2 ~ 3 " 5

1 3-2

+ 5^2 + 5^3 ( 1"¥>

-R,

where one can set aside every term (the subsequent parenthesis included), which follows the sign +.

R denotes the number of terms employed.

We obtain the better lower bound for N, when we aside those

terms, which multiplied by j - are less than the number of terms employed.

We give an example, choosing x = 1,000, D=l and p = 31

which is the greatest prime number not exceeding /x".

N(l, 103, 2, 3,....31) > 103 !-1.1 ...... J_ + J_ 2 3 31 3-2

+ _L + J_(1-1)+_L + _L(1-1 5-2 5-3 2 7-2 7-3 2

•li^TT^-IîT**1-?--

i l 13-2 13-3 (W)

+ W-F(W-4 + 7) 13-5

+ —!— + —L_ (i .1) + _J_ + _!_ ( 17-2 17.3 2 19-2 19-3

+ _L_ + _^(l-l)+-J_ + _ ^ ( 29-2 29-3 v V 31-2 31-3 v

We have set aside the term 17-5 (1

+ ( 1 -1-1 + -L) 7-5 2 3 3-2

+ 3 2>

3-2

. l , + - l - + -!_(i-l) 2 23.2 23-3 2 1, 52

1 1 2 " 3 " 3.2

) = 0.0039.

3 since 10 '0.0039... = 3.9... is less than 4, the number of terms

employed. In the term — 1 — (l - 1 - 1 - 1 + J _ + _i_ + _ L (1 -h) 11*7 V 2 3 5 3-2 5-2 5-3 2 >

we would at first set aside -}— (1 -]•) since — — (1 --) = 5*3 2 H'7-5'3 2

0.4... is less than 2, and we should also set aside the term

TT7 0 "1" 1" i + 3 2 + B ^ ) - 0 - 0 0 3 - - since 1o3-°-003- = 3. ... is less than 6.

We obtain then

N(l, 103, 2, 3,...,31) > 109 - 52 = 57 .

We can express this result in the following way:

When we efface among 1,000 numbers all the multiples of two,

three, five up to 31, there remain still at least 57 numbers.

Thence we deduce particularly that there exist more than 56 prime

numbers between 31 and 1000, on observing that

N(l, 103, 2, 3 31) = TT(103) - (/o 3) + 1

when we choose 0 as the starting point of the effacements.

Here IT(X) denotes the number of prime numbers not exceeding x.

Here we have chosen the domains u in a way to obtain the most

suitable lower bound. If we choose the domains u by the same

principle, we find

N(l, 103, 2, 3,...,31) > 109-52 = 57,

while TT(103) - TTC/IO3) = 158 ,

N ( l , 104, 2, 3 , . . . , 97 ) > 820-284 = 536 ,

while TT(104) - T T ( / K ? ) = 1,206 ,

N ( l , 105, 2, 3 313) > 5,733-1,862 = 3,871 ,

while TT(105) - it(A(F) = 9,528 .

In the sequel we w i l l choose the domains u by simpler

pr inc ip les .

108

To illustrate the principles sought after we give at first three

examples:

Eg. 1) N(l, x, 2, 3, 5, 7) > x 2 3 5 7 3-2 5-2

_L ( 1.1) + _ L + _ L ( i _ l ) + J _ 5-3 U 2] 7-2 7-3 K V 7-5

= x(i-l) 0 - i V-b v~b -24

C J D /

1-1-1, 2 3

3-2

16

We have set aside no terms.

Eg. 2) N(l, x, 2, 3, 5, 7, 11) > x 1 1 1 1 . J_ 2 " 3 " 5 " 7 " 11

+ J- + _L + _L o -1> + _L + -L (i -1) + -L (i -1 -1) 3-2 5-2 5-3 v V 7-2 7-3 v T 7-5 v 2 V

+ _1_ + _L_ (1 -1) + _L_ (1 -1 - 1) + _L_ (1 -1 - 1 - 1) 11-2 11-3 K r 11-5 U 2 3; 11-7 K 2 3 5;

- 26 ,

where the terms set aside are added on a small scale. One can also write

N(l, x, 2, 3, 5, 7, 11) > x

"(y

( i - ^ i - ^ i - ^ d - ^ d - ^ )

i + . _ J . _ + .. J _ . + .. J _ _ + •5-3-2 11-5-3-2 11-7-3-2 11-7.5-2 11-

+ ( I—)]. (1 + 5 + + 3 v M1.7-5.3.2/J V 1-2 1 -2-3 /

x[o.2078 - 0.0121 + 0.0004] - 26 = 0.1961x-26.

1

7-5-3J

Here we have set aside all terms of the form

of the form 1 PaPbPC

pd and

papbWe

Eg. 3) N ( l , x, 2, 3, 5, 7, 11 , 13, 17, 19) > x 1 1 1 2 " 3 " 5

7 11 13 17 19 3-2 5-2 5-3 * 21 7-2

+ _ L ( i . l ) + _ L ( 1 . l . l + _L) 7-3 2 7-5 2 3 3-2

+ - 1 - + J _ ( 1 . I , + _ 1 _ n-2 n-3 v r n-5

i - l - l 2 3

3-2

13-2 13-3 k r 13-5

1 1 2 " 3

3-2

1 1 1

+ - J - + _ ! _ ( ! - l , + _ i -17-2 17-3 2 17-5 \ i

+ — 3-2

2 3

19-2 19-3 v l' 19-5

0.163x - 72

Here we have set aside the terms on the right of the vertical lines. One see that the expression is of the form

Pa L p

aPb L L L PaPbPc p

aPbP cPd

where p , p. , p and p, run through the following values

p 2 3 5 7 11 13 17 19

pb 2 3 5 7

p 2 3 5 7 rc

Pd 2

in which a > b > c > d .

§3. We study at first the method employed for example 2.

We do not apply the general formula (5), but we deduce directly from the formula (3'):

N(D, x, p r...,pj = N(D, x) - I N(Dp , x) a < r

+ I I N(DPaPb> *> Pi»---»Pb_i) •

a < r b < a

On employing this formula twice, we obtain

N(D, x, p,,...,pj = N(D, x) - I N(Dp , x) a < r

+ I I N(Dpapb, x) a < r b < a

- I J 1. N(DPaPbPc x) a < r b < a c < b

+ 1 1 1 1 a < r b < a c I — + I I — a < r Pa a < r b<a PaPb

1 I I I a < r b<a c £ [ l - ^ + l2 - l3 - R

R ,

(7)

(7')

where £, is equal to the sum of the terms of the first of the

following three lines

Pi P2

Pi P2 1 +_U

+ Pl = a

(A) Pi p2

l? is equal to the sum of the terms formed by multiplication of every term of the first line by those terms of the second line, which lie on the left of this term, and J3 can be defined similarly.

We will say, in the sequel, that we calculate the expression

by means of diagram (A) or more briefly by means of the diagram

r terms > 1 three lines

We compare £o anc' a '• 2 2 2

° 2 = (IT-) + (ir) + ••• + (TT) + 2lz y 2\

or o^ > 2l2 .

112

We will also prove that

\ c < r K c ' \ a < r b < a Kapb /

> 3 ( 1 J ^ P F ) v a < r b<a c < b KaKbKc /

Any term , where y < 3 < a < r , is represented once in papgpy

£ 3 but, as we see, three times in a£2-

We search at first J- in £ — and —'— in I £ pa c < r pc P B P Y a £ r b < a

1 and then J- in Y -L and —!— in Y I ]

papb p$ c < r pc PaPY a < r b < a papb

and at last — in I — and ——- in £ £ . PY c < r pc PaP6 a < r b < a papb

The term is therefore represented three times in aj0 , rarS Y

which contains also terms of the form - = — etc. Hence we conclude

that olz > 3l3. P « P B

We can generalize the formula (7), on calculating the last sum

in (6) by means (6). On continuing we obtain a formula analogous

to (7) or more briefly analogous to (7'):

N (D, x, pr...,pr) > J [l - + l2 - ... - ln - R , (8)

where m is an odd number satisfying m<r, and where the expres

sion 1 - 1-. + Yo - ... - I is calculated by means of the diagram

r terms

- m lines

We can, in the special case m = r , calculate this expression:

i - I T + Z 2 - ••• + (-i)rIr

pl p2 pr

1 a < r pa a 5 r b < a papb

where r may be even or odd. The number of terms is 2 r in this

case. We obtain then the formula

In general case we will determine a lower bound for the

expression

We can, as before, prove that

whence am > mlj .

Hence we conclude

Lm m Lm-1

and

y < \ < (—) ^m m! \ m /

on applying the Stirling formula

, , m i! = ( f ) ( /2S+ e) ,

(1 £ (1 - Z-, + I2 - ... + H ) r Ir)

(L + 1 - L + ? + ... + ( - D r L ) f-m+2 - R

(9)

(10)

(11)

We know the value of the first parenthesis in the form of a

product. The second parenthesis is composed of a series of decreas

ing terms, whenever m + 2 > o, and then it has a value less than

lm+v which is less than ( _ ) .

We can therefore write v r I i m+1-

It is not difficult to determine the value of R

• • ' • ( K ) C < l + r + r 2 + ... + rm < rm+1

4).

We obtain then the formula

rm+l

(12)

whenever

m + 2 > o = -J-+...+-J- . pl pr

This formula is more useful than (9), the growth of r being

not so great as that of 2r. But the growth of the term R is still

too great for our purpose.

§4. For this reason we shall choose the domains u in another

way, setting aside all terms on the right of the vertical lines, as

in the example 3 (§2).

At first we set aside in the formula (3) all positive terms on

the right on a vertical line. We obtain then the following formula

See, for example, Landau: Handbuch der Lehre von der Verteilung der Primzahlen, I, p. 67.

N(D, x, P] pr) > N(D, x) - I N(Dpa, x)

+ I I N(DPaPh> x» Pi Ph.i) >

a < r b< a b < t

(13) where t is an integer less than r.

The terms of the last sum can be calculated by means of the same

formula, whence one deduces

N (D, x, P] pr) > N(D, x) - I N(Dpa, x) a< r

+ I I N(Dp pb, x) - I I I N(Dp p p x) a < r b<a a D a < r b<a c < b a D c

b < t - b < t c < t

+ I J lh 3 N(DPaVcpd' x ' p l Pd-1* • a < r b<a c 

-1 < e < 1 ,

1 " I — + I I —*— a < r p r a < r b<a papb

b< t

1

a < r b<a c < b â^bPc b < t c < t

+ 1 I I . .1 < r


1

a < r b<a c < b d < c PaPbPc^d b < t c < t d<u

(14)

N(D, x, p1 p r) 1 - S1 + S2 32n-l - R ,

(141)

where the expression

En = 1"V is calculated by means of the diagram in the form of stairs

En = l - S l + S 2 - ... - S ^

an a2 0]

J_ + ...+_!_+... +J_+... + _L_ _1 + ... + 1 Pi Pw-1 Pu Pt-1 Pt Pr

Pi pw-l pu pt-l

i.+ . . . + _ ! _ + . . . + J - + . . . + ]

Pi Pw-l pu pt-l

U... + ] pl Pw-l

u • 1 Pl Pw-1 '

We choose the prime numbers of the diagram as successive prime

numbers lying in the interior of the following intervals

-L _L- J- i R a n

Pl P"""1 ... P ° 2 P° P

r H Kr Kr Mr Hr

where a > 1.

We apply the Mertens' formulas, giving them the following forms: x -, c Y - = log logx+ 0.261 ... + e ^ — , -1 < e < 1

p 3 3 logx

S ( 1 - 1 ) = e 7 9 / 1°9 x °-561--- , -1 < 9 < 1 2 P logx

where log denotes the natural logarithm.

See, "Journal fur die reine und angewandte Mathematik" B.78, 1874, or Landau, Handbuch, I, p. 201.

Hence we conclude

f l - l 0 g a + e ! ! ^ , f(i-l)=le(1 + 1 ^ e / l o g x x p log x x P a

But in that case we can choose p, sufficiently large for which

a l =

a2 =

a n =

and

*1 =

TT2 =

\ =

whenever a Q

-L-+ ... Pt

* • • "

•L+... Pi

< - * >

* • %

* • %

> a .

+ p- < l 09 a 0 •

+ p ^ < log a0 ' •

+ pjl i < log aQ

- ( , T , > i ' • • • ( 1 ^ . , , > v

... (1- ' )> ] Pw-1 a0

(15)

(16)

We suppose particularly log C*Q < 1.

We try to realize a successive calculation of the sums, to which

the diagrams in the form of stairs give rise.

Suppose that we have calculated by means of the diagram

(2m-1) lines

giving rise to the expression E = 1 - S, + S^ - ... - S^m-] '

We subjoin then 2m+l lines on the left, (which only taken

gives rise to the expression 1 - + £ 2 - ... - £ 2 m + 1 ) :

1-Z1+Z2

^1

' ' ' *-2m+1

31

1 - s , * s 2 - - -

'

S2m-1

The sum J — is now equal to I, + S-,. We see also that the Pa ] ]

new sum H ——- is equal to £2 + S ^ + S2 on studying the three

possible cases:

p occurs on the left of L and p. on the left of L (£2)

p occurs on the left of L and pb on the right of L (S-|^)

p occurs on the right of L and p, on the right of L (S2) .

In general we can calculate the new expression E -| by the

following way:

Em+i - i - ( h + si) + (h + hh + h)-(h + sfo + hh + h)

+ ••• " fem+l + S 1 hm + ••• + S2m-1 k) '

We compare th i s expression with the fol lowing product

( W T + XZ- ••• * U «i -s1 + s2 - ... - s ^ )

= i - (Z-, + s,) + (Z2 + s1j1 + s2) - ...

" < W l +S1^2m+ ••• +S2m-1 h>

+ ^ 2 m + 2 + Sl W l + - " +S2m-1 W " - •

The first factor contains as many terms as possible, that is to

say, v is equal to the number of the terms in J\. The product

contains, as one sees, all the terms of Em+^ and in addition a

series of parentheses, whose values, by (10), are decreasing, since

H = CTm+l < 1°9 an < ' anc' n a v i n9 alternative signs. Hence we

conclude

Em+1 > V l Em " (E2m+2 + Sl Wl + - + S2m-1 I3> '

(17)

We can determine an upper bound for the last parenthesis. It is

a sum of the different products of (2m+2) numbers - , which all

occur in the two sums S, and £,. But we obtain the sum of all possible products of that form, on forming the sum

<S1 + ^ 2 m + 2

calculating by means of the diagram

?1

( 2 m + 2 ) lines

But by (11) and (15) we obtain

e(S1 + ^ ) \ 2 m + 2 /e (m + l ) loga 0 > 2m+2

e log aQ

2m+2

2m+2

2(m+l)

Our parenthesis (in (17)) is then still less, whence we conclude

that

Em+1 > V l Em " e log OQ 2m+2

We obtain then particularly, since E, = 1 - S,

E1 > 1 - log aQ ,

,4

Ep > TTp E, / elogaQ \ / / elogcu \

[—~ ) > 7 r 2 ( 1 " 1 ° 9 a O _ a o( -7 - j

on applying (16). On continuing in the same way, we obtain at last

 w2 Tr3 . . . i r M - l o g a 0 - a 0 '

n-l/ e 1°9 a0 a0

2 2n

or, since IT, < 1

whenever aQ

We obtain

E > 0.3(1 - — ) ... (1 - — ) (19) n v P^ v P/ l '

We study the number (R) of terms in E , on forming the follow

ing product

n__L_ ... . J_) M--L- ... -—!_) ... (i --1 -.... - — L ) . ( Pi Pr h Pt-1 Pi Pw-1

This product contains all the terms of E and more. The number (r+1) of terms in the first factor is less than p , and in

the second less than p 'a etc. We obtain the number of terms of i" 1

the product, on substituting all the terms — by +1 , whence we

conclude

R £ 0.3(1--^) ... (1-J-) - p* . (20)

This formula is valid for all successive prime numbers p-,,....pr

with p, > p , where p denotes a determinable prime number.

Suppose particularly p-j = Pe+1» the ( e + 1)" t n prime number.

When the question is to calculate N(D, x, 2 pe> p-],....Pr)»

we can subjoin to our diagram (under (14)) the following:

H* i

1 + 1 + 2 3

which gives rise to the expression

(i-l) ... o--L) = i-z1 + X2 * I P

containing 2e terms, whenever the number of the l ines are >e

We obtain then the new diagram

(e + 2n-1) l i nes

i - r ,+z 2 - ± i e

(2n-1) l ines

S2n-1 En = 1-S1 + S2--

giving r ise to the new expression En+1 :

En+1 = i - ( W + az+s,h+h> - •••

+ (Ie + s1Xe.1 + ... + se)

-(SiU + h U - i * - * ^ ) * -

* (S2 n .e Ie+ ••• +S2n-1 h > + •"• + (S2n-l U

or E „ , • ( ' - I 1 * I 2 - . . . * Z e ) ( l - S 1 * S 2 - . . . » S 2 n . ] )

• n - l ) . . . ti-i-iE, ,

where we have supposed e to be even.

We obtain then by means of (19) the formula

N(D,x,2,3 pr)> £0.3 ( 1 - ^ ( 1 - ^ ) . . . ( 1 . J _ ) . 2eP ^

(21)

valid for all r > e, where e denotes a determinable number, on

noting that every term of (1 -i)0 --~) ... (1 - — ) is multiplied

by every term of E . r

But in that case we can determine, by the Mertens1 formula, a

number c in a way that

N < D ' X ' 2 ' 3 Pr> > r&Tr - 2GPr < 2 2 >

for all r > c, where c denotes a determinable number (c > e).

If we choose D = l and P = p( /x~), i.e.; the greatest prime

number not exceeding /x~: p < /~x"L°^-2 ex \ / loqx

e 5/6 ^ , . > — • —

og x logx

for all x > XQ.

We can then state the following theorem:

When we efface from x consecutive numbers the terms from two

to two, then from three to three, etc; finally from p( /~x) to

p( /x~), there remain always more than terms, provided x>x n. log x u

The starting points of the effacements can be chosen as one would have it. xQ denotes a determinable number.

we can also deduce, by means of the formula (22), the following

theorem:

There exists always a number between n and n + /n , whose

number of prime factors does not exceed eleven whenever n > nn.

Choose in the formula (22)

D = l, x = /W and pr = p(n1 / n) .

We obtain then

N(l,/F, 2 , 3 , . . . , p ( n 1 / 1 1 ) ) > l ^ - 2 e n 5 / 1 1 > l log n

for all n > nQ.

When we efface in the interval [n, n + / n ] all the multiples

of two, three, etc. up to p ( n V H ) , there remains therefore at

least one number. We choose n as a starting point of the efface-

ments. The numbers not effaced cannot be composed of 12 or more

prime factors, because in that case one of these factors would be

less than / n + Jn , and therefore less than /n for all

n > nQ. But all these numbers being divisible by 2, 3,..., or

p(n1/]1) are effaced.

§5. We have supposed that

2, 3 pr

in the formula (21) are successive prime numbers.

We generalize easily on studying the non-successive prime

numbers

Vq2 Vl' Vl Vl' Vl qr forming a part of the successive prime numbers

Vq2 Vl' V Vl Vl' V VT*""qr ' where q-. = 2 etc;

124

and we obtain as before (see (21)):

N(°> x' ql Vl' Vl qrl

> ~ • 0.3(1 --J-) ... (1 -—!— )(1 --!—) ... (1 -J_) - 2eq5

or N(D, x, q1,...,qa_1, qa+1 qp)

(1 - — ) ... (1 - — ) x ql qr e 5

> - • 0.3 • — - 2 q . D i 1 r

(1-f) ... (1-f) Hence we conclude

N(D, x, q1,...,qa_1, qa+] qr)

> 0.168x ] 2e 5 D , 0 "vVf>. . . ( i - f>" r

We study now an arithmetical progression extended from 0 to x:

A A + D A + 2D

A and D being relatively prime. Suppose

n a c D = q ... q

We efface now the numbers being divisible by

qr---,qa-r qa+r---,Vi' Vi""",cV

on choosing q = q( /x~). We obtain

N<D' x» qi v r q«+i <v> > , , ; " 1 - 2 \ 0.168x „e_5

<f>(D)logqr

> 1.008x - 2e x5/6 > - 4

<j)(D)logx <KD) l o 9 x

fo r a l l x > Xg .

125

The numbers not effaced are indivisible by

q T " - ' q a - r qa+l qY-T V l qr

but they are also indivisible by

qa.-.-,qr .

since A and D are relatively prime. The numbers not effaced

contain therefore five or less prime factors.

Hence we deduce the following theorem analogous to that of

Dirichlet:

Every arithmetical progression, whose first term and difference

are relatively prime, contains an infinity of terms, whose number of

prime factors does not exceed five.

§6. Now we study the Merlin's sieve, where one efface double

all the multiples of three, five, etc. up to pr- On generalizing,

we study the following arithmetical progression

A A + D A + 2D

a1 a] + p] a] + 2p1

b1 b] + p1 b ]+2p 1

a a + p a + 2p r r Hr r Kr b b + p b + 2p r r Kr r rr

All the letters are defined in §2. Moreover we suppose a. b,.

and p, > 3 . Denote by

P(A, D, x, a r b r p r...,a r, br> pr)

or more briefly by

P(D, x, p r...,p r)

the number of the terms of the first progression, which are different

from all the terms of the other progressions. We deduce as before

the fundamental formula

P(A, x, a r b r p1 a r , b r , p r)

= P(A, D, x, a r b r p1 a r _ r b ^ , p r - ] )

- P(A', Dp r , x, a r b ] S p 1 , . . . , a r _ 1 , b r _ ] , p r - 1 )

- P(A", Dp r , x , a r b ^ p 1 , . . . , a r _ 1 , b ^ - j , p ^ ) ,


P(D, x, p 1 , . . . , p r ) = P(D, x, p 1 , . . . , p r _ 1 )

- 2P(Dpr, x, P p - . - . p ^ - , ) . (23)

It can give rise to no misunderstanding, since we have written

2P(Dp , x, p., pr_i) when one remembers that it denotes a sum of two expressions of the form P(A, Dp , x, a-,, b., p, ,...,a ,, b ,,

P M)-

We obtain as before, by means (23), the general formula

analogous to (5)

| p ( o . x , Pl p r )> i - i i . + n^-(i- i f) x ' r a < r pa ^ papb v c 3. Besides the designations, all are the same as in the formula (5).

We can also give the formula (24) the following form, on

supposing particularly p-,=3, p2 = 5, P3 = 7, etc.:

P(D, x, 3, 5 pr) > ±

+ J_ + J_ + J_(1-2) + 5-3 7-3 7-5 3

1-2 _ 2 3 5

Pr*? Pr'7

5-3

+ JL. + _L.(1.1, Pv.*3 Pv.'5 3

r rr

.1-1.1 3 " 5 " 7 4

\

5-3

+ JL + -L(1.1, / 7«3 7-5 V r '

R ,

(25)

where one can set aside every term, (the subsequent parenthesis

included), which follows the sign + .

We give an example, one studying the following arithmetical

progression extended from 0 to 11,776

1 3 5 7 9 11 13 15 11,769 11,771 11,773 11,775

[0 3 6 9 12 15

1 4 7 10 13

11,769 11,772 11,775

11,770 11,773 11,776

19

15

11,761

11,757 11,776

The starting points of the effacements are 0 and 11,776 (see §1).

We obtain by means of (25), on observing that a. =*= b., since

11,776 = 29-23 is indivisible by 3, 5, 7 19:

P(2, 11,776, 3, 5 19) > 11,776 £ _ £ _ £ _ .2 2 2_ 3 * 5 7 11 13 17

J_ + J- + -±_ + J_(1-£,+_i_ + _S_(1-i, 19 5-3 7-3 7-5 3 11-3 11-5 3

11-7

1-1-1 3 5 » + _ £ _ + _ 4 _ { 1 . 2 ) + 4

5^3

13-3 13-5 v 3' 13-7

+ -t_ + _i_(1.l,+_l. + _i_(1.l) _R , 17-3 17-5 v 3 19-3 19-5 v 3

where

R = 1 + 14 + 4 + 16 + 52 + 52 + 32 = 171 ,

1-1-1 3 5

5*3

whence P(2, 11,776, 3, 5,...,19) > 296 - 171 = 125 .

The number (t) not effaced of the first progression, whose number

is more than 125, having the following property: t and 11,776- t

are indivisible by 2, 3, 5,...,19. They cannot composed of three or

more prime factors, because otherwise one of these factors would be

less than Vl1,776 < 22.9 .

One can then write the number 11,776 as the sum of two numbers,

whose number of prime factors do not exceed 2, in 125 or more

different ways.

However, I have not succeeded in giving an example of the just

ness of the theorem of Goldbach by this method.

Nevertheless we see that we can deduce important results by

means of the formula (24), the method being completely analogous to

that employed above.

1 2 One should only replace — by — everywhere. pi pi

We calculate by means of the same diagram in the form of stairs l ?

as in §4 on replacing — by ~j- . One should then replace the P-j P.j

sums and the products considered in §4 by the following:

and °1 = P^+ • " + p ; < 2 1 ° 9 a 0 ' e t c - •

h Pr a 2

on applying the following formula

J (i - I ) = °-8322 . ece/iogx

3 P log2x

We suppose now 2 log ou < 1.

We deduce the following formula analogous to (18):

^1 > Vl Em " (e l09 " O ^ •

whence one gets

2 4 cu(elogou)

E n > n i - = „ p - Z l o g o , , - - ^ °— 1 -a0 (e loga0 )

Choose particularly

a = f = 1.25 and cu = 1.2501 . 4 0 We obtain then

E > 0.05(1-£-) . . . ( 1 - i - ) . (26)

We study the number (R) of terms in E , on forming the

following product

1-2 .... - Z V T . Z ..... 2 \2.. . A 2 ..... Pp Pr/\ Pi Pt-1/ V Pl "VI /

This product contains all the terms of E and more. The number (2r+l) of terms in the first factor is less than p when-

1/ot r

ever p-.>3s and in the second less than p ' , etc. Hence we conclude

2/ot 2/a" (a+l)/(a-l) 9 R < Pr P " ••• pr < pr " Pr •

We obtain then the formula

P(D. x, P] p r) > £ • 0.05(1 -j-) . . . (1 -j-) - p9r (27)

a formula which is valid for all successive prime numbers p, ,...,p

whenever p, > p , where p denotes a determinable prime number.

We obtain also a formula analogous to (21):

P(D. x, 3, 5 pr) > £ • 0.05(1 -|) ... (1 -j-) - 33p^ (28)

valid for all r> e.

Hence we conclude

P(D, x, 3., 5 p ) > * • ° - 0 4 1T - e

ep;? (29)

r D (log p r )2 r

for all r>c, where c>e.

Choose particularly p = p(x ). We obtain then

p(D.x, 3 . 5 . . . . . p ( x ' ' 1 0 ) ) > - ° ^ - 3 e x 9 ' 1 0 > °' 4 x

(30)

D(logx)^ D(logx)2

for all x> xQ .

On supposing D=l, we can therefore state the following

theorem:

When we efface double among x terms all the multiples of

ee, •

0.4x

three, five, etc. up to p(x ' ), there always remain more than

-=- terms provided x>x n. (logx)2 °

We have supposed that

ai * bi '

that is to say that none of the double effacements are reduced to a

single one. When the question is to determine the Goldbachian

partitions of the number x = 2 s p* ... p* , one see yet that ex y

a = b ,... ,a = b a a' * y Y

But the lower bound for P will not naturally less, when one

reduces the effacements (compare §5). One should then replace —

P: °

2 and — by

P

0.4x

D(logx)2

— . We obtain then the new lowe PY

pa py . 0.4 x

(1-f) ••• O-h D(logx)2

a

Hence we conclude, as in the previous example, on choosing

D = 2 , the following theorem, analogous to that of Goldbach:

One can write even number x, greater than xQ, as a sum of

two numbers, whose numbers of prime factors do not exceed nine. XQ

denotes a determinable number and the prime factors can be different

or not.

We can also deduce the following theorem:

There exists an infinity of the pairs of numbers, having the

difference 2, in the class of the numbers whose numbers of prime

factors do not exceed nine.

§7. We can also determine an upper bound for the number of

numbers, which remain non-effaced on employing the sieves of

Eratosthenes and Merlin.

We apply the following inequality

N(A, D, x, a-|, p-j,..

< N(A, D, x, a.|, p-|,

or more briefly

N(D, x, pr...,pr,..

where r<n.

• ' V pr V pn }

•••»ar, pr)

•,Pn) < N(D, x, p r . (31)

We apply also the formula

N(D, x, p r . . . , p r ) = N(D, x) - I N(Dpa, x) a< r

+ I I N(DP,Pb ' P i ' - - - 'Ph-V a < r b<a a D ' D '

(3 ' )

To estimate the terms of the last sum, we apply (31) and the

same formula (3'). On continuing we obtain the formula analogous

to (14):

N(D, x, pn pr) < JV

I I I a < r b<a c<b papbpc

b< r c< t

1 - X — + I I — a < r p a a < r b<a papb

b < r

1

1 a < r b<a c<b d<c papbpcpd

b < r c < t d < t

or more briefly

D(D, x, p1 pr) <

where the expression

l-s, + s 2 - ... + s 2 n

+ R , (32)

+ R ,

En = 1 - S] + S2 .. + S 2n

is calculated by means of the diagram

On employing the same method as before, we obtain

2m+3

Em+1 < V l Em + e 1 og ag

and par t i cu la r l y

whence

E l < n l +

E2 < n-|n2

elogot

e l o g o o \ 3 ^ e l o g a ~5

1 + a o — r - + ao

On continuing we obtain at las t

E < n....n n 1 n

, t «0,'J^)\sjt^ 'f +

or

E n < ( , " ^ ' - < , - ^ | l *

e l o g a .

1 - a r

e 1 og a0

/ e l o g c u \ whenever OIQ I J < 1 .

Choose par t i cu la r l y

a = — and ou = 1.51

We obtain then

E„ < 1.505(1 - -L) . . . (1 - - L ) . n v Pi Pr

We study the number (R) of terms in En on forming the

fol lowing product

1--L- ... - J - ) i - f Pl p r / \ Pi

1

Pt-1

_1_

Pi " •Vl

We see, as before, that

R < p J p 2 / a . . . p S / a n < p i 2 a / a - 1 ) = p! •

We can give (32) the fol lowing form

N(D, x , P l pp) < J • 1.505(1 -J - ) . . . ( l -J-) + P6r •

Thence we conclude the formula

N(D, x, 2, 3 pr)

< ^ . 1.505 ( l - l ) ( l - l ) . . . ( l - l - ) + 2ep6r

va l id for a l l r > e.

But in v i r tue of the Mertens' formula we obtain

N(D, x, 2, 3 , . . . , p ) < ° ' 9 X + 2ep^ r D log p r r

fo r a l l r > c , where c > e .

Choose par t i cu la r l y p = p(2 /~x") . Thence we conclude that

7 /T XQ.

On applying the inequality (31), we obtain

N(l, x, 2 p(/T))< N(l, x, 2 p(6/x-))

< N(l, x, Z p(2 7/7)) < 7x

(34)

logx

for al1 x > xQ .

Thence we conclude particularly that

ir(x) - ir(/x)+ 1 < - ^ - . logx

whence

w(x) < T ^ - + /x< 8x

logx logx

for all *>x Q, TT(X) denoting the number of the prime numbers not

exceeding x.

On comparing the theorem in §4, we obtain also

x < N(l, x, 2,...,p( 6/x"))< T^- . (35) logx logx

When we efface among x terms all the multiples of two, three

iys I

, 7x

etc. up to p( /x") , there remain always N terms, where N is a

number lying in the interval , whenever x>x Q. .logx logx

We study at last the sieve of Merlin. We obtain the formula

analogous to (33): 2 3

2 2 / an(e loga 0 )

n p1 p r \ 1 - a 2 ( e l o g a 0 ) '

Choose par t i cu la r l y

a = 1.

whence one gets

a = 1.25 and aQ = 1.2501

E„< 1.82(1-1-,... ( 1 - A ) .

Thence we deduce as before

P(D, x, 3, 5 , . . . , p r )

< £ . 1.82(1-|)(1-|) ... (1 - f ) + 3epJ°

or

P(D, x, 3, 5 , . . . , p J < ] - 6 x 2 + 3ep™ . (36)

r D(log p r )Z r

for all r>c, where c > e (see §6).

Choose now p = p(2x ' ). We obtain then

P(D, x, 2, 3 p(2x1/]1)

< 194x + 3 e+10 x 10 / l l < 195x

D( logx) 2 D( logx) 2

fo r a l l x> x„ .

We apply now the inequality

P(D, x, 2, 3,...,p(/x"))< P(D, x, 2, 3 p(2 U/x))

and the equation

Z(x) - Z(^T+ 2) + 1 = P(2, x, 2, 3,...,p(/x) ) ,

where Z(x) denotes the number of the twin prime numbers not exceeding x, and where we have chosen 0 and 2 as starting points of the effacements.

We obtain therefore

Z(x) < 195X , + Sx +2 2(logx)z

or 100 x

Z(x) < (logx)2

where Z(x) denotes the number of the twin prime numbers not exceeding x for all x>x 0, where xQ denotes a determinable number. Here

(See: Skr. Norske Vid.-Akad; Kristiania, I (1920) no. 3. Some formulas in the text are slightly modified by the Editor).

Translated by Yu Run Bui

New Improvements in the Method of the Sieve of Eratosthenes

A. A. Buchstab

In 1919, V. Brun [l] gave a method for applying the sieve of

Eratosthenes to a series of problems in number theory.

V. Brun proved that there exists infinitely many integer pairs

such that 1) each integer of the pair has at most nine prime

factors, and 2) the difference of integers in each pair is equal to

2, where 2 can be replaced by any given even number.

V. Brun established also that every large even number is the

sum of two numbers each having at most nine prime factors. In 1924,

Rademacher [2] improved the number nine to seven in the above

results. In 1930, I was able to improve the number seven to six,

and it was also established by Estermann in 1932.

In this paper, I give a new approach to these problems in which

the number of prime factors is decreased to five. By the use of

precise iteration of integrals, the number of prime factors can be

decreased further.

We consider the problems of solubility of the equations

2 = n' - n" and 2N = n'+n", where the number of prime factors of

n' and n" is required to be bounded by a constant. The other

problems considered by Brun and his successors can be treated

similarly.

At the same time, I obtain a better upper estimation for the

number of solutions of the above equations.

The estimations given in the first and second lemmas are obtained

by the ordinary Brun's method.

As distinct from the other works, I shall obtain here the

sufficiently closed upper and lower estimations. The basic part of

our work was introduced in my paper [3] .

1. We use P (x , y) to denote the number of non-negative

integers < x such that they are not contained in any of the 2 r + l

progressions

a 0 , aQ + p0> aQ + 2 p 0 , . . .

a l ' a l + p l ' a l + 2 p i " " *

b-j. b1 + p p b1 + 2 p 1 5 . . .

(1)

a r , a r + p r , a r + 2 p r , . . .

b r , b r + p r , b r + 2 p r , . . .

where PQ = 2 , 0 < a Q < 2 , p. the odd prime number < y wi th the

order 3 = p, < . . . < p r < y , 0 < a . 9 8 ^ i log2x

holds uniformly on u the set of a. and b i , where c is a

constant.

According to Rademacher's paper c i ted above, we have

where

E = ( 1 " 21

l f , + , l 1 I P ^ 0 " 2 , l K ^ l < a < r K a l < a < r l < b < r , KaKb 1 < c < b Kc

I I I 1 l < a < r l < b < r , 1 < c < r, 1 < d< r2

papbpcpd

(1-2 I f ) + ... 1 < e < d Ke

and R denotes the number of terms —— in E. pa ph"-

1/10 Let r=r, and p be the greatest prime number < x .

Let p„ be the greatest prime < y}/(1°w ~ ) f0r 2 < k < t + l , rk

where B = —- - e and h= /e - e in which e denotes a pre-assigned

positive number and t is the integer such that p ' < o)n < p rt+l u rt+l

When t+1 < k < n, we take p = p and p = pn = 2. rk rt+l rn+l u

We denote by E. for k = l,2,...,n (En = E) the sum of those

terms in E, where their denominators have only prime factors with

indexes greater than r.+,, i.e., the denominators have at most

2k+1 prime factors. Hence

E - 1 - E<]) + + E<2k) - E ( 2 k + 1 ) tk i tk + ... + tk tk

where E V ' denotes the sum of those terms in Ek with exactly i

prime factors in their denominators.

Let S/1' be the i-th elementary symmetric function of the

numbers

and

2 2 2 > n n

w1 Vi + 2 \ r k 2

nk = n (l - f ) k i = r k + i + 1 P i

Then we have evidently

where

Ek+1 >- h nk+l - *k+l • ( 2 )

(2k+4) (2k+3) (1) (3) (2k+l) $ k + l = S k + l + S k + 1 Ek + - " + S k + l E k

*. . , = 0 i f k+1 > t , (3)

and

E > n - s<4>

By the successive application of the inequalities (2)

we have

E = En > h n 2 ... nn (i - J- s{4> - h - *3

Choose mQ such that

S^1) = 2 I — < 2 log — < 0.516 , 1 a = r 2 + l Pa 17

sj1} =2 I -J- < 2 log Vi" = 1 (2 < k< t) a = r k + 1 + l P a 2

S s £ n ) < — < - J — ( l < i<2k+ l and 2 < k < t )

K " i ! 21 i !

and in particular,

s d ) 4 s\^ < — < 0.003

1 4! Therefore

r l -1 2 f = n ( l - f ) < ( f f ) < 1.675 , n l i = r2+l p i 17

1 ^ 2 _1

' = n O-f) < /e (2<k<t) . nk i = r k + 1 + l Pi

Denote E*1' by

E(1) . cd ) + s ( i " 1 ) E^1) + +S ( 1 ) E( i_1> + E ( i )

h ~ \ + bk h k - l ••• bk Lk-1 + bk- l

( i = l , 2 , . . . , 2 k + l , E ^ = E<^+1> = 0) .

Since

E,(i) = S,(l) < 2.14 • -1- (i = 1,2,3) , 1 ' 41

we have for all k < t-1,

E(i)<2.14.4e

2(k-1) , K 41

and therefore from (3),

$k+l < 2 J 4 , - 2 P 4 e 2 ( k _ 1 ) (e2_5) (2*k+1^) Hence

$ = ^ $ 2 + n 2 k $ 3 + - - < 2 - 1 4 ( e 2 _ 5 )

. 1 (eV2 + e6/2 1 + e l l / Z 1 + . X 4

6 V 4Z 44 /

= 2.14(e2-5)e1 / 2 4 - 6 (l - ) < 0.0087

and

1 - J_ (s{4) + $) > 0.98 .

Denote A = lim (2 log log x+ I log(l - | ) ) .

x+« \ 3< p< x v / P When n is suff iciently large, we have

n n = n (l -£) = 100 • - ^ - + o(—!—) ,

3<p<x 1 / 1 0 l 0 g X l 0 9 X

i .e . ,

E > 98 -JS— + 0(—L-) log^x logJx

where c = 1 eA = 0.4161 ...

R does not exceed the number of terms —— in the papb'--

expression

"1 o \ / "l , \ / "2 (.-.U)(.-.i.^-i.fu.-.f2 b = 1 p b / \ c = l P c / \ d= 1 Pd

and thus

R < (2r, + l)3(2r,+ l ) 2 ... (2r + 1 ) 2 < p3 p2 ... p2

- \ 1 / \ 2 ' v n ' Kr-, Kr„ Kr I l n

< A x3/10 x2/10B x2/10Bh _ = A x(3/10)+ 2h/(10B(h-l))

Since j - + -—^—- is less than 0.999 if e is sufficiently

small, we have

R < 0(x0-999) = 0( x ^

And therefore log3x

P (x, x1/10) > 98 -£*=_ + o{*-) u log^x log^x

Lemma 2.

P (x, x 1 / 1 0 ) < 101.6 — ~ - + 0 ( -2L_ ) log 'x log°x

By the same designation, we have for 3 = p, < . . . r , > r 9 > . . . . Let p be the greatest prime i c r-|

< x ' , i . e . , r = r,. Let p„ denote the greatest prime k

<xl/(10Bhk"2) f0r 2 < k < t + l , where B = | | - e and h = V e " - e

in which t denotes the integer such that p < uv, < p rt+l ° rt+l

When t+ l<k<m, we define p = p and p = 2. Then rk r t+ l rm+l

Ek = 1 - E k 1 ) + E k 2 ) " - - - + E k 2 k ) 0<k<t)t Em = E

and

Ek+1 K- nk+ l Ek + *k+ l »

where (2k+3) (2k+2) (1) (3) (2k)

*k+l = Sk+1 + Sk +1

Ek + ••• + Sk+1 Ek

•k+1 = 0 if k>t

and (3)

E ] < n] + s1

By the successive use of the above inequalities, we have

E = E <n,...n M + j - s|3) + — — <k0 + „ J „ $, + ...) m l m \ n ^ l n-|n2

2 n1n2n3 3 / Choose u)n such that

r l s { ] > = 2 I -L < 2 log | | < 0.2453 ,

a = r2+l pa "

s ( l )3 sj3^ < — < 0.0025 ,

' 6

r l , ,-1

I J*-ft ' ( I ) '1 -2 8 • n l a = r2+l v Ha

and the estimations of sf1 ' and — are similar to those given K nk

in Lemma 1 for k>2. Now we evaluate the $• as follows.

*2 = S<5) + S<

4) E<]) + S|3) E{2) < 0.00154 ,

E ^ - S J ^ + EJ 1^ 0.746 ,

E ( 2 ) = S ( 2 ) + S ( 1 ) E ( D + E ( 2 ) < 0 2 7 8 ^

144

E ( 3 > = S<3> + S<2) E^) + S< ] ) E f ) <0.068 ,

E<4> = S<4> + S|3) E^) + S f ) E<2) < 0.012 ,

and from E^1^ < 4.45 • \ ( i =1,2,3,4), we have c 41

E^) < 4.45 e 2 ( k _ 2 ) — ( i = 1,2,...,2k)

and

*k+1 < 4 - 4 5 e 2 ( k _ 2 ) ^ 7 3 ( e 2 - 5 ) •

Therefore

112113 3 112113114 $ = —— $0 + — • $, + ... < 4.45(e - 5)

. 1 + ^ 5 ~ + /e^" +

4 4 4

2 -7 / e5/2 \_1

4.45(e - 5)e« 4 ' [1-- ) < 0.0074 "16 and

1 + -(s! 3 ) + — $, + $)< 1.016 . n-| V 1 n 2 2 /

By the definition of m, n, ... n is also equal to

n ( i - | ) = 1 0 0 - ^ - + o ( — L _ ) . 3<p<x 1 / 1 0 P l 0 9 x l 0 9 x

For suff iciently small e, the estimation of R is given by

R < A 2 x 2 / 1 0 x 2 / 1 0 B x 2 / 1 0 B h . . .

= 1/5(1 + h / B ( h - l ) ) < A 0.9999 . 0 ( _ x _ ) , c L log3x

and therefore we have

P (x, x 1 / 1 0 ) < 101.6 S±— + 0(—x- ^ w log2x log3x

145

Where c is the constant defined in Lemma 1.

Lemma 3. Let u and v be two constants or variables depend

ing on x such that 2 < u < v < A , where A is a constant. Then

I 1

x 1/ v<p<x 1/ u p(log^) 2

= 1 / log *4 + JL - * W 0(-L_) . 1og3x V «-l "-1 v-1 ) S o g V

The proof is similar to the proof of a lemma in paper [3].

2. Consider now the function P (x, x ' a) (a<10). It is

evident that for 2<a<10,

Pu(x. x1'0) < Pu(x. x

1 / 1 0) , (5)

PJx, x1/a) > 0 . (5a)

By Lemma 1 and inequality (5), it asserts that there exists a non-

decreasing function X(a) which is continuous or has only a dis

continuity of the first kind on the interval 2<a<10 such that

PJx. x1/a) > x(a) - ™ - + 0 ( - V ) u log^x logJx

holds uniformly on u, where c is defined in Lemmas 1 and 2.

For example, \[a) is defined by

\(a) = 0 , if 2 < a < 10 ,

\(a) = 98 , if a = 10 .

By Lemma 2 and (5), it follows that there exists a continuous

non-decreasing function A(o) on the interval 2 < a < 10 such

that for any u, we have

P (x, x1/tx) < A(a) - ^ - + 0 ( - ^ - ) . u log^x logJx

As an example, we may take

A(a) = 101.6, if 2 < a < 10 .

We use A.(a) and A.(a) to denote the functions with the

properties as X(a) and A(a).

Theorem 1. Suppose that A.(a) and Ak(a) are two functions

satisfying the above conditions. Then the function y(a) defined

by ^(a) = 0 , if 2 < a < x ,

fP_1 z+1 Y ( a ) = A . ( f 5 ) - 2 A . ( z ) - ^ i d z , i f 3<x<a<10 ,

1 Ja-1 K z 2

where 3 is any number satisfying a < 8 < 1 0 , is also a A-function

i.e., V(o) = X i + 1(a).

First, notice that the difference between Pjjx, Pr+i) and

P (x, p ) is equal to the number of integers < x in the progres

sions

ar » ar + p r , ar + 2pr,...

br , t>r + pr , br + 2pr,...

but not in any of the first 2r-l progressions of (1). If a + kp

/ x - a \ = a. + np., then k = a! (mod p i ) , i > 0 ( 0 < k <

r , a! < pi J,

and if a r + k p r = b ^ n p ^ then k = b' (mod p..), i>l (b| < p..).

Therefore the number of integers < x in the progression a ,

a +p , a +2p ,... but not in any of the first 2r - 1 progres-r r r x - a

sions of (1) equals to the number of non-negative ingegers < r

which are not contained in the progressions Pr

a: I a! + pi , a! + 2p.,..., 1=0,1 r-1

b! , bj + pi , bi! + 2pi,..., i =1,2,...,r-1

x - a,

~r \ *Y

/ x - a x i.e., it is equal to P , 1 — — — , p 1, where u' denotes the set

of integers al and b! .

Similarly, the number of integers in the progression b ,

b +p , b +2p ,... but not in any of the first 2r-l progressions

of (1) is equal to P ,, ( -y1-, pr j . P^, ( —^, pp j is coincident

with P , {-j*-, p ) or they are distinct by 1, and the same situa-r V P r 7 / x - b

tion holds for P „ "V

We have

(V-"0 •"" p-t(t-^ ' p„<x- PrH> ' P J * ' "r> - p»' (¥-' pr> " P«'<f • »r> " \ •

r r r r (6)

where 0 Prfl) and PJX« x V 8 ) = V x > Pt> •

By the successive use of (6 ) , we have

^ / c u = D /» v ] / 3 P„(x. x 1 / o ) - P„(x. x 1 / B ) - I P . ' ( f ' P i )

x l / B $ P i < x l / « W i V P i ^

1 _ l / aP - i ( ^ P l ) - Z w 1 ' x 1 / B < P i < x "

where £) j . = 0( /x~) .

For s imp l i c i t y , we omit the index i hereafter. Let

6 - CL

u = a + - s-1 (s = 0,l n) , s n

(7)

where c, log x < n < c? log x. For given prime number p with

l / ( u s + 1 + D V ( u + 1 ) < V(u . + 1 +1) l / ( u + l )

x < p < x

log p/(1og x / p ^ T)

l / ( u s + 1 + l ) l / ( u + l ) x s ' < p < x s

1 i / ( V i ) ( P a ) i X p , ( X p )

< p< x / x x log p/( log x / p k

+ P a ) » ( p ' ( p ) )

/K < 2 c I - Ak

l / ( u s + 1 ) V(u s+1) , 2x k \ logp x < p< x P

+ 0 I X

V ( u s + 1 + 1 ) l / ( u s + l ) , 3 ; . X SP<X P 9 P

and by Lemma 3 ,

/ „

+ 0 | log

Therefore

g 3 x V 9 us us V l / )

x 1 / 6 < p < x |<x , / ,(p"i (f-"*'-! (f"")

=0 log 'x s=0 K 5 \ us s s+l /

+ 0 logJx s

n " ' / u „ , i u „ , n - u \ \

s=0 V us us u s + l / /

Since

n-1

s=0

,3-1

V I u . . , u

I A(uc+1)hog-lt l + -^L_l u u u ,

s s s+1

dzx A.(z)(dlogz + ^ ) + 0(£) a-1 K zl n

and n-1 / u , u , - u

we have

T < 2c

and from (7 ) ,

log^x

-1 z+1 A (z) £2 - dz + 0(—*=-) a-1 K z^ log3x

Pu(x, x 1 / a ) > _ £ l _ ( x . ( 8 ) - 2 log^x \ J

rS-1

a-1 A|r(z) *£dz )+0(-

^ ' log^x

i . e . ,

\ - + 1 ( a ) = X.(3) - 2 6 - 1 z+l

Ak(z) ^ d z a-1 k zl

The theorem is proved.

Theorem 2. Let X.(a) and A. (a) be two functions satisfying

the stated conditions. Then the function ui(a) defined by

fB_1 z+l u(a) = A. (P) - 2 x,(z)^-dz (3 < a < 10) ,

K Ja-1 1 zZ

where 3 is any number sat is fy ing a < 3 < 1 0 , is also a A- funct ion,

i . e . , u(o) = A k + ] ( a ) .

The proof of the theorem is completely similar to that of

Theorem 1.

Let 3 < 10. Let AQ(a) and AQ(a) be two functions on the

interval 6 $ a < B such that

XQ(OL) = 0, if ct< 3

AQ(B) equals to a positive constant ,

and

AQ(a) = A0(g), if 6 < a<

By the successive application of Theorems 1 and 2 and starting

from the functions with index i =0, we have

X.-1+1(x) = y s ) - 2 a-1 1 y2

-1 x+l A i + 1 ( 6 - l ) = A 0 ( B ) - 2 J e _ i X . + 1 ( x ) ^ d x

-1 x+l = AQ(B) - 2AQ(e) I ^± dx u u J3-2 x2

fB-1 fB-l x+, v+1 + 4A.(3-1) 2± ^ - d x d y (1 = 0,1,...) ,

1 J3-2 Jx-1 x2 y2

where for all i, A i(B)=A Q(6) and A i(6)=X 0(6), and

A^y) = A^B-1) if y<6-l.

The coefficient of A.(3-1) in the right hand side of the

above expression is less than 1 if 8 < 6 < 10, and thus by the

successive iteration, we may obtain a A(3-l) which is sufficiently

close to the root of the corresponding equation, i.e., we obtain a

A-function AQ(a) defined by

151

A0(6) - 2A0(6) 3-1 x+1

An(a) = An(6-1) = 3-2 x2

dx

+ e r3-l r3-l

1 -4 f'i^L.y^dxdy Jx-1 xd "2 J3-2 Jx-1 x- y

where 7<a<3-l and e is a sufficiently small number.

Now we obtain a new A-function AQ(a) defined by

AQ(3-1) = A0(3) - 2 3-1

6-2

v+l

A n ( x ) ^ - d x - e

= AQ(3) - A0(3-l) 6-1 x+1

3-2 y& '

An(a) = 0 if a < 3-1 .

By the s imi lar method, we may obtain L. and AQ from AQ

and AQ, and so on. Start ing from AQ(10) = 98 and AQ(10) =101.6,

the resul ts are given as fo l lows.

A0(9) =

101.6-2-98 [ — dx h x2

+ e = 85.1

1 - 4 x+1 y+1 . , —y • i-y dx dy

x-1 yc y6

An(9) = 98 - 2-85.1 ^0

x+1 ^ - d x - e , = 75.58

8 x^ '

85.1 - 2-75.58

AQ(8) =

8 x2+l dx

7 x< + e' = 72.86

1 - 4 ° x+1 y+1

x-1 x y dxdy

L ( 8 ) = 75.58 - 2-72.86 8 x+1

i l L d x - e ' = 53.51 , 7 xd '

72.86 - 2-53.51 7x+l .

A0(7) = — + e" = 67.58

x-l x^ yc 1 - 4

and finally

X(6) = AQ(8) - 2AQ(7) 7x+l Z ± dx - eV = 0.03 5 xd '

i.e.

and

P (x, x 1 / 6) > 0.03 S* + 0 ( — V ) (8) u ' ' log2x log3x

P(x, x 1 / 6) < P (x, x 1 / 7) < 67.58 - 2 L _ + 0 (—*=-) w w logzx log3x

hold uniformly on w.

In particular, if we take a. = 0 and b. = p. - 2 then P (x, x^'6) is equal to the number of integers n<x such that n and n + 2 do not divide by any prime number < x 1 / 6, i.e., there exists a pair of integers n and n + 2 each having at most 5 prime factors.

The inequality (8) shows that such pairs form an infinite set.

That is, we have proved the following result.

There exists infinitely many integer pairs such that each integer

of the pair has at most 5 prime factors and the difference of integers

in each pair is 2.

Suppose that x is an even integer, a. =0 and b. is the

least non-negative residue of x modulo p.. Then a. = b. if

p.|x. In all the estimations of P (x, x'' a), the factor — £ = -1 log x

should be replaced by — ^ - v(x), where v(x) = n -j^- . Since log^x p|x P"

p>2 ¥ ( J ) - v ( x , t 2 - v ( x , ( 1 + 0 ( ^ - ) ) i f x 1 7 1 0 < P < x l / 2 , the

results corresponding to Theorems 1 and 2 still hold, and we have.

There exists a constant AQ such that every even integer

greater than AQ can be represented as the sum of two integers

each having at most five prime factors.

Concerning the number of prime pairs Z(x) in the interval

[2, x] , we have

Z(x) < 2 8 . 2 — * — . log x

fo r x > x Q , here c = -= -< 0.417 is used.

References

[1]. V. Brun, Skr. Norske Vid.-Akad. Kristiania, I, No. 3 (1920).

[2]. H. Rademacher, Abh. Math. Sem. Univ. Hamburq, 3 (1924) 12-30.

[3]. A. A. Buchstab, Mat. Sbornik, 44 (1937) 1239-1246.

(See Mat. Sbornik, 46 (1938) 375-387.


On Prime Divisors of Polynomials

P. Kuhn

We use small Latin letters to denote the natural numbers which

are also called numbers for simplicity, and p, q, s, t the prime

numbers. Let

Pn(x) = aQxn + a ix

n - 1 + ... + an , aQ > 0 (1)

be an integral-valued and primitive polynomial which is a product of

r (l<r<n) integral-valued, primitive and irreducible polynomials. bl be The fixed divisor of (1) is assumed to be T = t, ... t .

We shall find out a possibly smaller integer k such that there

are infinitely many numbers in

P n 0 ) . Pn(2),...,Pn(x),... (2)

which have at most k prime factors besides the fixed prime divisors

ti (1 < 1 < e ) .

H. Rademacher [l] and G. Ricci [2] have used the Brun's sieve

method to treat this problem, and they have found out a smaller

number k for the case r=l only. It needs also to determine a

corresponding smaller number k for the case l < r < n . Here the

proof for the case r = n will be given.

Suppose that p runs over all prime numbers satisfying

P < x 1 / v , p + t1 (3)

where v will be determined in the latter as a function of n.

Let d be an integer satisfying d£0(mod p) and d£0(mod t.).

Let N (dx, x ' v ) be the number of integers in (2) which are n b.+l

-°°,

Nn(x, x 1 / v ) > Cn • 0 .98xv n l og " n x

0 ( x x ( n ) / v v n + l l o g - n - l ^ n

+

Nn(dx, x 1 / v ) < Cn • 1.016 H Vn l o g A

+ 0( x * ( " > / v vn + 1 l og " " " 1 x) ,

X(2) > 9.99 , X(3) > 13.67 ,

X(4) > 17.50 , X(5) > 22 .02 , . . . (4)

where C is a positive constant depending on P (x).

Theorem. If r = n, then it is sufficient to take k = u+n,

where w is the smallest integer such that

^ - (<u+l) > nlogx(n) . (5)

For examples, n = 2, k = 6; n = 3, k = 10; n = 4, k = 15; n = 5,

k = 21

Suppose that v=2x(n) in (3) and that q runs over all prime

numbers satisfying

V 1 / V < q < (2a0x)J + 1 , q * t. . (6)

Let m denote the number in (2) satisfying

m < Pn(x) , m £ 0(mod p) ,

b.+l 2

mSOCmodt,1 ) , m £ 0(mod q ) . (7)

Let U be a lower estimation for Mfx, x / v) the number of

integers m. Since Mn(x, x1 / v) = Nn(x, x

1 / v) + 0 ( x 1 - 1 / v ) , we

have by (4)

+ 0(x1/2 log"""1 x) + 0(x]- 1 / v) . (8)

U = Cn • 0.98xv" log"nx

Let M (qx, x ) be the number of m which is E 0(mod q). Let

V be an upper estimation of £ M (qx, x''v) = L , where q runs

over (6). Then L < £ N (qx, x'' v), and therefore by (4), we have n q n

V = c • 1.016 I H vn log"" x+ 0(x log"""1 x) . (9) q q

If m has at least w+1 prime factors q, then it is counted

at least w+1 times in L , and therefore the difference n

A = U V gives a lower estimation for the number of m which OJ+1

has at most w prime factors q and its other prime factors are

s > (2aQx)2+l besides those t.. The number of such prime factors

s is at most n when x-*-°°, since each linear factor of P (x)

has at most one prime factor s. Hence if

0.98 («+!)> I - , (10)

1.016 q q

then A-*-°° when x-*•<». Since £ — ^ log x(n)» the theorem is A q q

proved.

References

[11. H. Rademacher, Abh. Math. Sem. Univ. Hamburg, 3 (1924) 12-30.

[2l. G. Ricci, I, Ann. Scuola Norm. Sup. Pisa, (2) 6 (1937) 71-90; II, Ibid. 91-116.

[3]. P. Kuhn, Den Skand. 12-th Mat. Kongr; Lund (1953) 160-168.

(See Proc. Intern. Cong. Math.; Amsterdam, 2 (1954) 35-37.)


Noted by the Editor: Let N be an even integer. Let Po(x) =

x(N-x) (x = 1,2,... ,N-1). Then by the similar arguments, one

can prove (a, b), where a + b < 6.

DET KONGELIGE NORSKE VIDENSKABERS SELSKAB FORHANDLINGER BD XIX, NR 18

511.287

On an Elementary Method in the Theory of Primes

By

ATLE SELBERG

(Innsendt til Generalsekretaeren 18de oktober 1946 av herr S. Selberg)

In the following we shall give a brief sketch of an elementary method which can be applied to the same problems as the "sieve method" which has been developed by BRUN. The new method is very simple and also generally yields better results than Brun's.

Suppose that we have given a finite sequence of integers a the total number of which is N. We shall study the problem of finding an upper bound for the number Nz of a's which are not divisible [1] by any prime < z .

We define a sequence of numbers X for 1 < y < z such that Xt = 1 while the other X's are arbitrary real numbers. Then obviously

a y/a T , , 7 J < * T, 72

/a X

where x denotes the greatest common divisor of ji and y2 • We now suppose that when p is a positive integer it is possible to find an

approximate formula for the number of a's which are divisible by p, in the form

y i = — N + R P

where R. is a remainder term. We further suppose that the function f (p)

Atle Selberg: On an elementary method in the theory of primes. 65

function f(p) is multiplicative, that is, that f{PiP2)~f(Pi)f(P2) if 1

Px and p2 are coprime integers. Since can be said to represent the

'probability' that a is divisible by p , the latter supposition means that the 'event' that P i /a shall be independent of the 'event' that p 2 / a , if Pi and p2 are coprime. If this is the case we have

I =-r-3L_JV + fl, = ^ — N + R Tf j , „ / 7 i 7 2 \ 2l2l_ f(yi)f(72) ^ L

la

Inserting this in the above inequality for N^ we obtain

7 l , 7 , < 2 A(7l) /172) 7 , .7 2 <2

Writing now

* 7 , X 7 .

7 l , 7 , <z r (7 i ) '172)

we shall determine the X 's for 2 < 7 < « such that Q becomes a minimum.

We write when p is a positive integer,

MP) = lM(d)/f4")

where ju(d) is MOB 1 us'function, in particular if p is quadratfrei

hw-m\)k'w) According to a well-known formula we then have

fix) = Ifi(p) = I A(P) , P/* P/7,

P/7,

D. K. N. V. S. Forth. XIX, 1946. B 5

66 D. K. N. V. S. Forhandlinger BdXIX, Nr 18

Inserting this in the expression for Q we get

Q = I fi(p) p <, z

I Ply fit)

y <.z

Writing for 1 < p < z ,

I Ply fti) y < z

we have that

W) ]T n(p)yt

z p < —

y

py

Now we determine the minimum of the form

p < z

under the condition

X M(P)yp = Xi

= 1. P<z rd)

We easily find that the values of y which make the form a minimum are

P-(p) 1 yP =

flip)

p' < z

M V )

fi(p')

and that the minimum value of the form Q is

I p < z

1

U-HP)

fl(p)

For the corresponding values of the A 's we find for 1 < y < z,

Atle Selberg: On an elementary method in the theory of primes. 67

X = 7

f(y)

(2)

v?(p) p<z fl(P)

=M(T) n i l P/N

p< — y

M(P)M(P7)

ft(py)

f{p) y M2(P) p<z fl{P)

X z p < — y

(P.7> = 1

P-2(P)

A ( P )

Inserting these values of the X's in (1), we obtain

N N2<

(3) P-2(P)

+

I p<z fAP)

7 i . 7 2 < ^ ' ~ Y "

which, if the last term on the right-hand side is not too great, give us the

required upper bound for Nz.

Applying this method to the number a = n(n + 2), 1 < n <s x we find, _L _

taking z = x 2 where e is a sufficiently small positive fixed number,

that the number of prime twins not exceeding x is less than

10.6x

log2* -, x>x0

This is better than the best upper bound established by BRUN'S methods.

I have also developed a method based on similar principles for dealing

with the problem of a lower bound. A full account of these methods, with

applications to several problems, will appear later elsewhere.

[1] We may replace this condition by a ^ rp (modp) for all primes p < z,

where the r ' s are integers depending on p only.

Trykt lste februar 1947 1 kommisjon hos F. Bruns Bokhandel

Aktletrykkerlet i Trondhjem

SCIENCE RECORD New Ser. Vol. I. No. 5, 1957

MATHEMATICS

ON THE REPRESENTATION OF LARGE EVEN NUMBER AS A SUM OF TWO ALMOST-PRIMES* t

WANG YUAN (3£ x )

Institute of Mathematics, Academia Sinica {Communicated by Prof, Hua, L, K.t Member of Academia Sinica)

For the sake of briefness, we write the following proposition by (a, b).

Every sufficiently large even integer can be represented as a sum of two integers > 1, of which one contains at most a and other at most b prime factors.

The aim of the present note is to prove (3, 3) and (a, b) (a + b <; 5) by the method used in previous papers11-2l. These results improve the (3, 4)[sl of the auther in 1955. Moreover, using ByxmraS's1*1 method with more complicated numerical calculations, we have (2,3). Recently, we have found some mistakes in numerical calculations in the proof of (3, 3) of A. H. BuHorpaflOB16'. We shall state it at the end of this note.

In this paper, p denotes prime number and pt denotes *-th odd prime.

Let x be an even integer and £ be a real number. Let (w) a; at, b{ (1 < i < r) be a sequence of integers satisfying the following conditions: (1) « = 0 o r l ; <)<«<, b{ < pit iipjx, then a{ — bt, otherwise a,. =£ bk

( l < . - < r ) , where p, <: f < ptjrX. Let P„ (x, £) be the number of integers n satisfying the following conditions: (2) 1 < » < x, n==a (mod 2), n # at (mod pt), n # 6{ (mod pty

(1 < * < r). Let v > « > 1 be two given positive numbers. Let S{ denote the set

of integers n (x - n) satisfying the following conditions:

(3) 1 < » < * , n (x - n) # 0 (mod 2), n (* - n) # 0 (mod^) (1 < * < « ) -

where p, < x^~ < /> Let 2K denote the set of integers n(x — n)

* Received June 29. 19S7. t An integer Is called almost prime, if the number of its prime factors is not exceeding

a fixed constant.

292

satisfying (3) and the following conditions:

<4) n (x-n) # o (mod#+J.) 1 < ;' < i s ,

where pt ^ x ~~»~ < p We denote the number of elements of R and

i J . -1_ 2R by N (x, x "»") and M (x, x •, x « ) respectively.

Lemma 1. M (x, x~,x~*~)=N [x, x~) + 0 (*1-_ir) + 0(*"=").

Proof. N{x, x-T) - M(*,*"£",*"=")< J - 2 | 2 1== * ' • <P< X " i < ; B < j r

»(**») = <) (mod/*)

* • </><# •

(i) #1*: S , < 2 l - [ j ] + l -0(*^h. « = o (mod £)

(ii) fl* Sf = 2 l < a [ ^ ] + 2. n (*—n) = o (mod p»)

Hence, we have

— 0 ( * ' - ^ ) + 0(x^~).

Thus we have the lemma.

Lemma 2. There exist sequences of integers (w,) (1 <^; <C i—s) satisfying (1), such that the number of elements in 3R with at least I prime

l _l_

divisors in the interval x' <.p ^.x « is not exceeding

* i < / « - * r,+' W

Proof. For l<;y<l<—s, let Tj be the subset of 3JI whose elements are divided by p,+j~ Indeed, T,- is the set of integers n (x—n) satisfying (3), (4) and the following condition:

(5) *(*-») =s 0 (mod £+,).

(i) A+i I * : Evidently, the number of elements of fy is not exceeding

1) If n— x—n' and n (x—n) (= n' (x—n')) belongs to gj or (R, then we agree that »(*—n) be computed twice.

293

2 i-o(^,_"h. »=o(mod *>,+,)

(ii) P^j\x: From condition (5), we deduce that n = o (mod pl+i) or •» == z (mod ^>4+;) Let

{w/j a — 1; ii pf / x, then a< — 6,. — 0, otherwise «,• = 0,

6,.&+, = * (mod£) ( l < « - < 5 ) .

Evidently, the number of elements of TV is not exceeding

If n(x-n) belongs to 2Ji and n(x—n) has at least I prime divisors in JL JL

-the interval x » < ^ ^ * « , then n {x - n) belongs to at least / different sets Tf Hence we have the lemma.

Lemma 3. Let C > 1 be a given number. Then there exist two non-decreasing and non-negative functions X (a) and A (a) (o < a C) with the properties that each of A. (a) and A (a) has at most finite discontinuities, such that the following inequality holds uniformly in (w) and z

<•> X^To^+ Q( l 0 gyio g lo gJ<P-^^)<M^) 1^V+

+ Q ( i o g y i o g i o g , > o < z < c ) '

-where c, = 2 ev TT (1- .. 1 2 J TT^jg and 7 is Euler's constant. * » 2 /> |*

p>2 The lemma follows immediately by Bran's method. Fundamental Theorem. Let m be a non-negative integer and C >v>

ti> 1 be three positive numbers. Let X (a) and A (<*) (0 < a ^ C) be two iunctions with the properties as stated in lemma 3. If

then for all sufficiently large x, there is always an integer n in the inter--val 1 < n <x-l, such that n{x-n) has not prime divisor less than or

equal to x~*~ and has at most m prime divisors in the interval af»~ 1, 2, . . -.

294

1 JL

Then we have N (x, x » )= P$ (x, x » ). By lemma 1, 2 and 3, for sufficiently large x, the number of elements in 2J1 with at most m prime divisors in the interval x » <p^x <• is not less than

1 1 1 —_. T> I x _i \ . i

M(x,x• ,x«) — s r r r X w> \j^,.x~v'r °(* ")

= Pi (x, x±) - InTT 2 Pw; ( - ^ , * ~) + 0 (*h + 0(*1 ~h l<j<t-s

~2> (»«-sri:>&)^)^* log2*

+ " ( •

c , * » )

> 3 . log2 x log log;

This means that for sufficiently large x, there exists in the interval 1 < n

•Hx—1 an integer n having not prime divisor <laf»~ and having at most

tn prime divisors in the interval x~*~ <fi^. x~*~. Thus we have the Theorem.

By Brun - 5yxniTa6 - Selberg's method (C/[3]), we have the following table:

(8) a

A (a)

A (a)

...

8

68.52S11

60.88817 ...

6

43.0082

26.70925

...

...

...

S

34.89666

9.18109 ...

4

29.39023

»

...

...

and

From table (8), we deduce that

*(6>- T S X - O T ) T * * > °-33829

^8> - T I X I T T ) ^~dz > °-56125-Hence, (3, 3) and (a, b) ( a + 6 ^ 5 ) follow from the Fundamental theorem.

By Byxiirrafi's method141 with more complicated numerical calculations, we have improved the values in table (8) and obtain the following

(9) a

A (a)

A (a)

...

...

...

8

64.403149

63.S9931

...

...

...

7

S0.S29826

47.471252

...

...

...

6

41.01897

31.004145

...

...

...

5

34.89666

13.61559

From table (9), we have

295

^-u:*(^-)-^-">"» 7

Hence, (2, 3) follows from the Fundamental Theorem. Finally, we indicate some mistakes in the proof of (3, 3) of A. H. BH-

HorpaflOB*. Here, we use his notations and omit the explanations. He has obtained that (10) 3.2 7 -2IX< 0.3167.

Hence for sufficiently large z, we have

(11) j iX

i 8 ( « ) ( 2 « + l ) i « < - ^ ^ ( 3 . 2 7 - 2 7 1 ) + 0 ( ^ I = 1 ) < 0.5.

On the other hand, we know that 8 (w) is a non-negative and non-increasing function and 1 —8 (2) = (4.5—4 log 2) e'". Hence, it follows that

(12) CX 8 ( « ) ( 2 « + l ) d « > r s ( w ) ( 2 « + l ) ( * K > 8 ( 2 ) r ( 2 M + 1 ) * « > 1 . 5 , J I.I J 1.1 J l I

which is a contradiction with (11). Moreover, one can prove

(13) (4. ip _ 4 £ w T i M _ . ( 2 « + l)<*«< 16.81 -

_ 4 f a S M <2u+l)du<-l. J i . » i - e ( « ) v ;

Corresponding results of twin-primes problems have also been obtained.

REFERENCES

[1] Wang, Y., 1957, On Sieve Methods and Some of the Related Problems, Science Record* Academia Sinica, New Ser. I, 1, 9—12.

[2] Wang, Y., 1957, On Sieve Methods and Their Some Applications, Science Record, Academia Sinica, New Ser. I, 3. 1—6.

[3] Wang, Y., 19S6. On the Representation of Large Even Integer as a Sum of a Product of at most 3 Primes and a Product of at most 4 Primes, Acta Math. Sinica, 8, 500-513.

[4] EyxniTa6, A. A., 1940. O PasJioaceHHH HSTHUX Hncea H» Cymty ,u>yx Cnarae-HUX c OrpaHOTeHHira HHCJIOM MHOJKHTeJieii, flAH CCCP; 29, 544—548.

[5] BHHorpaAOB, A. H., 1957. IlpHMeHeHHe i (s) K PemeTy 3pa-roc<l>eHa, .name.*, eS; 41: 1 pp. 49—80.

* Added 6. Sept. 1957. I have been informed that this mistake has also been discovered and corrected by himself (Cf. Ma-reit. c6. 41 (83): 3, 415, 1957).

Reprinted from Collected Papers, Atle Selberg, Vol. 2. © 1991 Spring- Verlag

45. Lectures on sieves

A. Selberg

Dedicated to the memory of Viggo Bran

Contents

1. Introduction 66

2. The sifting problem 66

3. More specific assumptions 70

4. Further assumptions, definitions and main objects 74

5. Remarks concerning remainder terms 75

6. Some general principles for constructing, combining

and modifying sieves 77

7. Preliminary results using A2 and A2 A" sieves 82

8. General theory and existence theorems 88

9. Remarks on the preceding section and

generalizations 108

10. Some asymptotic formulas relating to the A2 and

A2A~ sieves 117

11. The combinatorial sieve as developed by Brun,

Buchstab and Rosser 127

12. Remarks on the combinatorial sieve, possible

generalizations 153

13. A study of sieves in connection with a

particular simple sifting problem 155

14. The shifting limit for constant sifting density 165

15. The effects of relaxing the conditions on the R^

in our sifting problem 180

16. Two examples in connection with the B2R method 183

17. Some upper bounds for sifting limits for

constant sifting density 192

18. A historical digression, the parity principle and a

further example 202

19. Sifting on an interval, the uses of Fourier analysis 208

20. An extremal problem with application to the upper

bound sieve for the interval and a digression

on the large sieve and Hilbert's inequality 213

21. Some historical comments on the last section 225

22. Remarks on the Brun-Titchmarsh theorem 226

23. An early approach to the twin prime and the

Goldbauch problem 233

Bibliography 246

23 . A n ear ly approach t o t h e tw in prime and t h e Go ldbach problem

We shall for simplicity confine ourselves to the twin prime problem, and so avoid the complication of dealing with a general even difference or even sum. This is a complication of detail only, and it is easily seen how one would modify the approach for the problem of a general even difference or even sum. The result we shall give is by now of historical interest only since after the appearance of E. Bombieri's theorem on primes in arithmetic progressions, one has been able to treat these problems as problems with sifting density one

234 Atle Selberg

rather than two, and with the additional ingenious ideas brought in by Jing-Run Chen results have been obtained that go far beyond what the approach here described could be expected to yield even with further refinements.

As an illustration of the variety of ways in which the A2 method may be adapted to a particular problem, it remains however of some interest. The reader may notice the affinity with the ideas in my attempt of 1946 to prove the existence of primes in the interval (x, (l-f-c)x), which was briefly described in Section 18. The proof here dates from late 1950 or early 1951, the result was first proved about a year earlier using a set of weights similar to those used in [Ankeny 1] to prove the result.

The method could equally well have been applied to the Goldbach problem, but to avoid having to pay special attention to the prime factors of the number which we wish to show can be represented as a sum or difference of primes or at least "almost primes" (numbers with very few prime factors), we choose here the twin prime problem.

We shall need a series of lemmas.

Lemma 16. Let k be a positive integer, r(n) be the divisor function and

Dk(x)= £ r ( n ) ,

n<x

then we have (23.1)

w , , . ( ^ ( * , + e + J S ^ ) + o ( * n ( I + ^ " ) . Here, if k is not squarefree the sum and the product taken over the p\k on the right hand side are extended only over distinct prime factors of k.

It is of course enough to prove this for squarefree k. Lemma 12 gives

(23.2) D(x) = Di[x) = x (logx + c) + 0(y/x).

It is easily seen that

(23.3) Dk(x) = £ MMth)D (•£-) . d,\k \dxdiJ d2\k

Expressing D(x/(did2)) by (23.2) (which holds uniformly for 0 < x < oo) and inserting in (23.3), we get the remainder term directly, while the main term requires a little manipulation to get (23.1).

We shall need some facts about Kloosterman sums:

(23.4) S(m,n;k)= £ ^ = ^ 6 3 hh=l(k)

63 H.D. Kloosterman [1].

Lectures on sieves 235

where m and n are integers and h runs over a complete set of residue classes relatively prime to k.

We have

(23.5) \S(m, n;k)\< (m, n, k)1/2k1/2T{k) .M

Lemma 17. If either m or n is = 0 (mod A;), we have

(23.6) gKn;*)-5<m'*l5fr0;*>-0,

w/ii/e t/ m and n are 6o#i noi = 0 (mod k) we have

(23.6') S(m,n;k) — S(m,0;k)S(n,0;k)

<2(m,n,k)1/2k1'*T(k).

(23.6) here follows from 5(0,0; k) = tp{k), while (23.6') follows easily from (23.5) and the fact that

S(m,0;k) = £ n{k/6)6, S\(jn,k)

|S(m,0;Jfc)| <<p(m,Jfc).

We define for \a\ < |7r,

(23.7) 5« (a )= £ e2™"1, 0<i/<t

and have then the inequality

(23.8) IS(a)| < 2|a|

L e m m a 18. For (k,£) = 1 and k < x2 we have for any e > 0, that

(23.9) Z?M(x)= £ r ( " ) n=* (*)

IF* fl0gx + c+2j : -^)+O(A:- l /4 x l /2+£^

64 The deepest part of this (fc = p) is due to A. Weil [1], the rest then follows from the results of H. Salie [1] (for k = p r , r > 1) and the multiplicative properties of Kloosterman sums.

236 Atle Selberg

We have the two identities (23.10)

f s M W ? = £ r{n) logX-

n<x

- * 1 E . "*«r*(5)^(i)7--k/2<m,n<k/2

and (23.10')

f o k ( ( ) 7 = E r W l o g ^ (n,fc)=l

n<x

= i £ S(m,0;fc)S(n,O;fc) f S t ( = f ) S l / t ( £ ) f . * -k/2<m,n<lt/2 •" V * ' V * 7 £

These identities are easily proved by inserting the expressions for the Kloost-erman sums and the St(m/k) Sx/t(n/k) and carrying out the summations over m and n.

Observing that when (k,£) = 1, we have S(n,0;k) = S{£n,Q;k), we get by dividing the second identity with <p(k) and subtracting it from the first, that by Lemma 17, (23.11)

I lx ~ / %dt 1 / • * , , < &

\LD^7-lMk)iD^l <l E (m,n,k)^k^r(k)jf-JXj

k 0<|m|>|<*/2 4 l m n l Jl *

= 2fc1/2r(fc) logx E ( m ' n ' f c ) 1 / 2 < Gk^2r(k) log2(fc + 1) logx 0<m,n<k/2 m n

where we have used that

(m,n,kY'2< E d 1 / 2 , d\m d\n d\k

so that

„ (m,n,fc)^2 < y ^ d i / 2

0<m,n<fc/2 d|fc

\

E ± < E d " 3/2

\0<m<*/2 /

< <(3/2) log2(fc + 1) < 31og2(fc + 1).

0<m<*/2


Using now the inequality valid for 0 < 6 < 1,

as well as the formula for £>*(<) given in Lemma 16, we get from (23.11) that

^ W = ^ x f l 0 g x + C + 2 E ^ ) + o ( ^ x l o g x )

+ O Qfc1/2r(fc) log2(fc + 1) logs) .

Choosing here the 8 which makes the two remainder terms of the same order we get the result of the Lemma, with a sharper but more complicated remainder term.

Lemma 19. We have for (6k,I) = 1 and 6k squarefree, (23.12)

£ 2"W = d x n - f ^ r (log* + C + 2 £ i ^ +0(fc-VV/»+«), n=/(6t) p\kP[-P i> \ p]kP ZJ

for any £ > 0, here u(n) is the number of prime factors of n, counted with multiplicity, and

Let bjL= (l-p->)>= ( 1

^ d « Al i _ 2 p - 1 1 ^ p»(p>-2)

We see that 6 > 0, and that the first singularity on the real axis is a simple pole at s = 1/2, thus we have

£ - ^ = 0(iogx).

d<x v d

now for (n,6) = 1, we have

2"w = £ M V d ) , d\n

which gives

£ 2"<"> = £ &d £ *-(">) = £ bdD^x/d), n<i m<x/d

238 Atle Selberg

where dd= 1 (6k). Inserting here the expression for

D6k,u(x/d)

from Lemma 18, and summing the series

J2 4 and Yl 4logd, (d,fc)=l ° (d,k)=l a

we obtain the result of the Lemma after some simplification.

Lemma 20. / / f\ and f% are multiplicative functions and oij and u)2 are additive functions on the positive integers, then for n squarefree and A a constant, we have that

(23.14) £ h(u)f2(v) (A + Wl(«) + wj(t/)) = / s(n) (A + wj(n)), uv=n

ui/iere / j is multiplicative with

(23.15) h(p) = flip) + flip),

and u3 additive with

(23.15') /s(p)ws(p) = / I (P)WI(P) + f2(p)"2(p) •

This is simplest verified by considering for variable s the identity

eAs £ /i(u)/2(t/)e»<*"<">+"»<''» = e*s I I ( / I ( P ) e""lCp) + A W e" 2 ( P ) ) -uu=n p | n

forming the derivative with respect to s and putting s = 0.

Lemma 21. / / / is multiplicative and w additive, we have for squarefree d

(23.16) /(dMd) = £/ '(p)u,V),

/iere u/(p) is an additive function defined by

(23.17) / , ( P K ( P ) = / ( P M P ) .

This follows by considering /,(d) = /(d) e*"(d) and writing

/*(<*) = £ / » P i d

where f'Jjp) = /3(p) — 1. Taking the derivative with respect to s and putting s = 0, we get the result of the Lemma.


We shall denote by hk(v,w) the function defined by (10.11) in the case of constant sifting density k, and have then

Lemma 22. We have for v > w,

(23.18) * ( „ , „ ) = _ _ ( _ ) ,

where -y is Euler's constant.

From the form of Lemma 7 it is clear that for constant sifting density k we have that for v > w

hk(v,w) = ck(—j ,

to determine the constant ot, we put w = 1 and consider for v > 1

c. = , % K l ) = - / e - / o . * - = — j e * k •

—too —ioo

Here we may write

rvs \ _ g _ ( roc g~* / <i£ = 7 + log us + / — dt

JO t Jvs t

and get

Letting here v —* oo, we get

Ck~î J ?+» r(Jfc+l)'

e ioo

*7 T „» p - * 7

-ioo

which proves the Lemma.

We now consider the two expressions

(23.19) Qi=QiW= E {2"(n) + 2"<n+2>}x J E A 4 -n = - l ( 6 ) |,d|n(Ti+2) J l < n < 2 i

and

(23.20) Q2 = Q2(A)= E J E A 4 • n = - l ( 6 ) (d\n(n+2) ) l < n < 2 l

240 Atle Selberg

For Q2 we have immediately

where f(d) = d/r(d), and K = (di,d2). To obtain a similar expression for Qi, we have first to evaluate

(23.21) Nd= J2 {2^n) + 2^n+i)} = Nd + N'd\ d|n(n+2) n = - l (6) i < n < 2 i

where

(23.21') N'd = Yl 2" ( n ) ' d\n(n+2) n = - l (6) x<n<2x

and N'J is a similar expression with 2J/ '"+2 ' instead of 2"'"'. We write

(23.22) Nd = £ T V ^ d\d2—d

where TV' = y^ 2^" ' = 21/'<il) V^ 2"'m '

rfll" x/di<m<2x/di d2 |n+2 m=£(6<*2)

n = - l (6) x<n<2x

where £ is the residue class mod6d2 determined by d\(. = —1 (6), dt£ = —2 (c^)- Since (^,6^2) = 1, we have immediately from Lemma 19, that

We may now evaluate N'd from (23.22) and (23.23) by using Lemma 20 with fi(p) = 2, w,(p) = - logp, }2{j>) = ^ 2 and u;2(p) = ^ , which gives

p - 1

and . , 2 p - 4 , 2 , 4 logp

W3(p) = - 3 p ^ l l 0 g P = - 3 l 0 g P + 3 3 p ^ -


From this we get

^=c•sfl?^(lc*I+c"-5,ogd+5S¥s^)+0<d",/v,^",

for e > 0. For N^ we get the same expression so finally we have

(^*.«,£nfc«(1,,+c.-|b,,+|j:it) + 0(<r1 /V< / 2 + £) .

Writing / 4 for the multiplicative function with

/ 4 ( P ) = - 3 ^ T '

and 0J4 for the additive function with

4 logp w4(p) = 5 : 3 3p - 4 '

we now get (23.25)

+0 (••"".£. OT1^1

We now try to make the ratio of Q1/Q2 as small as possible. Since this is a rather intractable minimum problem, we shall instead just choose our A's so as to minimize

with Aj = 1. This is the standard problem. We find

(23.26) Q1(min) = Kr- = - L ,

and

/4M L , ^ /i(p)

242 Atle Selberg

Since fA corresponds to constant sifting density 3, we get from Lemma 22 and Lemma 7 that

(23.28) E . - ? ^

= £!l n i x n md-V e aii, (i - w 3ii/4(p)V ?

= — n — i — x n (p"1)4

6 - 2 7 ^ ( l - l / p ) 3 3<V<2p2(p-2)2 i o g

3 z n ( p - i r

^ 2 - 9 % 1>

1 3 P 2 ( P - 2 ) 2

= 2C12log3z.

Since we have that with our choice of A's,

/4(«l)/4(rf2)

equals zero for any tj(d) with w(l) = 0, it only remains to evaluate

(23.29) £ £ g ^ (* log«-«*(*)),

by the use of Lemma 21, we can write

'2 /«(«) (-log*-<*(*)) = £ /i(pV(p)

P I <*l p\d-2

and so transform (23.29) to

Here, since u/(p) = §logp + 0((logp)/p), we may replace u/(p) by f logp without changing the asymptotic behavior. The resulting sum can then be handled by Lemma 7' in much the same way as we handled 5IZ, w e get that

V "'(Pi ^ 2 , - 1

S^)~Cilogz-Combining our results, we get that the main term in (23.25) is

J_ logz + §logz

~C[X (log zf •


Since our |Ad| < 1, the remainder term is

thus if we choose z = x1/3~c, we have

(23.30) Q l ( A ) ^ _ x _ _ _ _ .

We next turn to Q2(A), we have

We first wish to evaluate the expression in the curly bracket. From (23.27) we get

Xd_ _ lMf*(d) v _ 1

f(d) £* /(d) £ K(a) '

thus

(23.32) ETT5T = ^ - £ ^ W ^ !

PI<* /W E, pfT\. ' M ffc)

Here we first turn our attention to the summation over d with p and a fixed. We have then

v ii(d)Md) _ fi(p)Mp) v /x(tf)/4(g)

, f r , /(d) /(/>) «£/, /(«) = M(P)/4(P) n f , W P ) \

,1ii*-4,ii,*'-4

, p - 2 V 3p - 4

Inserting this in (23.32) we get

£ * PIP P 2 P \ a p \ a P 2

a<z

= M P ) TT 2 ( P - 1 ) v n i

E* p | p ( P - 2 ) 2 . ' t ' / p p l i ' P - 2 ' (,r',p)=l

244 Atle Selberg

Here the last sum can be evaluated easily by standard elementary means, we get it is

= « \ n ^ ' ° V ° ( n ( ^ ) ) . Inserting this in (23.33) we get

(23.34) E 7 ^ = , ( p ) ^ n r ^ l o g £ + o ( - L l M n ( 1 + M \

fa / (d) E* p | p P - 2 P \2lz P P |„V VP/J

Going back to (23.31) with (23.34) and inserting the main term we get

(23-35) '$£m*r The sum occurring here is again easily evaluated by Lemma 7' we find it is asymptotic to

^ Q l o g 4 * .

Using also the value of J2Z from (23.28) we get finally (since the remainder term in (23.34) is seen to give only a contribution of lower order) that

(23.36) Q2(\) 4 d log2 z

Combining this with (23.30) we see that

(2337) « L W ^ 4 ^ + 1 N

( 3 3 7 ) Q*W A(logz + 2)>

since z can be taken as xll3~e with any positive £ we can bring this ratio as near to 14 as we like. This shows that we must have some n for which

(23.38) 2"(n) + 2" (n+2) < 16 ,

and since

d log" z

we see that the number of such n must be at least of order x

log i

But (23.38) means that of n and n + 2, one has at most 2, the other at most 3 prime factors.

It is possible to reduce the ratio (23.37) somewhat, since we are not actually at the minimum of the ratio. By adjusting the Ad's suitably Gerd Hofmeis-ter was able to bring the ratio down from about 14 (which is what we get if


we put z = x1?3, so we can come arbitrarily close to 14) to about 13. This is not enough to improve the result qualitatively, for that one would need to bring it down below 12 (since 12 = 4 + 8), but it is enough to exclude for instance the case v(n) = 3, v{n + 2) = 2, this could be done by replacing the expression

2"(") + 2"(n+2)

by

1 2"(n) + - 2" ( n + 2 )

3 3 in Qi(A). This would not affect the ratio Q1/Q2, but one sees easily that if the ratio is brought below 40/3, this excludes the case v(n) = 3, u(n + 2) = 2 without permitting any new possibility. The bound for our z essentially came about since the remainder term in (23.9) makes that formula effective only for k < x2?3~e. If one could prove that the main term dominates sufficiently (by a certain power of log x would be enough) for k < xa one could choose 2 as xa/2~c and would end up with 8 / Q + 2 + e instead of 14 + e, thus a > 4/5 would eliminate the possibility that of n and n + 2 one has 2 and the other 3 prime factors. Of course better results could be obtained for a smaller a if we choose our A's so as to get nearer the real minimum of the ratio Qi/<?2-Another possibility of improvement is to replace the expression

d\n(n+2)

in Qi and Q2 with

61 |n, Si\n+2

where the Aj,^ are not just dependent on the product 6^2, but can be chosen freely except that Ai,i = 1. Since we can estimate N'd d and N'd\d%

well enough as long as did2 and d\ d2 both are < z 2 ' 3 - £ , this means that we need to assume A^,^ = 0 only for

max(<51152/3,<5i/3(52) > x 1 / 3 ' £ ,

instead of for <5i<52 > i 1 / 3 _ £ . We thus have the choice of about x2/ ,5-£ A's instead of i 1 / 3 _ £ . This is much more difficult to handle, but should lead to better results. These would not however be as sharp as Jing-Run Chen's result in any case.

246 Atle Selberg

Bibliography

(Essentially lists only contributions believed to be somewhat relevant to the contents of these lectures)

Ankeny, N.C., Applications of the sieve. Proc. Sympos. Pure Math. vol. VIII, Amer. Math. Soc. Providence, RI 1965 pp. 113-118

Ankeny, N.C. and Onishi, H., The general sieve. Acta Arith. 10 (1964/65) pp. 31-62

Bombieri, E., On the large sieve. Mathematika 12 (1965) pp. 201-225 Bombieri, E., The asymptotic sieve. Rend. Accad. Naz. XL(5) 1975/87 pp. 243-

269 Brun, V., Le crible d 'Eratosthene etle theoreme de Goldbach. Christiania Vidensk.

Selsk. Skr. (1920) Nr 3, 36 pages Buchstab, A. A., New improvements in the method of the sieve of Eratosthenes.

Math. Sbornik (N.S.) 4(46) (1938) pp. 375-387 Chen, J. R., On the representation of a large even integer as a sum of a prime

and the product of at most two primes. Sci. Sinica 16 (1973) pp. 157-176 Chen, J. R., On the distribution of almost primes in an interval. Sci. Sinica 18

(1975) pp. 611-627 Diamond, H.,G., An elementary proof of the prime number theorem with a re

mainder term. Inventiones Math. 1 (1970) pp. 199-258 Iwaniec, H., Rossers sieve. Acta Arith. 36 (1980) pp. 171-202 Iwaniec, H., The half-dimensional sieve. Acta Arith. 29 (1976) pp. 69-95 Iwaniec, H., Almost primes represented by quadratic polynomials. Inventiones

Math. 47 (1978) pp. 171-182 Jurkat, W. B. and Richert, H. E., An improvement in Selberg's sieve method. I.

Acta Arith. 11 (1965) pp. 217-240 Kloosterman, H. D., On the representation of a number in the form ax2 + by2 +

cz2 +dt2. Acta Math. 49 (1926) pp. 407-464 Kuhn, P., Zur Viggo Brunschen Siebmethode, I. Norske Vid. Selsk. Forh. Trond-

hjem 14 (1941) No. 39, pp. 145-148 Landau, E., Uber die Einteilung der positiven ganzen Zahlen in vier Klassen

nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. d. Math. u. Physik (3) 13 (1908) pp. 305-312

Montgomery, H., The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978) pp. 547-567

Rademacher, H., Beitrage zur Viggo Brunschen Methode in der Zahlentheorie. Abh. math. Sem. Univ. Hamburg 3 (1924) pp. 12-30

Salie, H., Uber die Kloostermanschen Summen S(u,v;q). Math. Zeitschr. 34 (1931) pp. 91-109

Salie, H., Zur Abschdtzung der Fourier-koeffizienten ganzer Modulformen. Math. Zeitscher 36 (1932) pp. 263-278

Selberg, A., On an elementary method in the theory of primes. Norske Vid. Selsk. Forh. Trondhjem 19 (1947) no. 18, pp. 64-67

Selberg, A., On elementary methods in prime number theory. C. R. 11 Skand. Math. Kong. Trondheim 1949 pp. 13-22

Selberg, A., The general sieve method and its place in prime number theory. Proc. Internat. Congr. Math. Cambridge Mass. 1950, vol. 1, pp. 286-292

Selberg, A., Sieve Methods. Proc. Sympos. Pure Math. vol. XX Amer. Math. Soc. Providence, RI, 1971 pp. 311-351


Selberg, A., Remarks on Sieves. Proc. 1972 Number Theory Conf. Univ. Colorado, Boulder, CO, pp. 205-216

Selberg, A., Remarks on multiplicative functions. Springer Lecture Notes, vol. 626, pp. 232-241

Selberg, A., Sifting problems, sifting density and sieves. Number theory, trace formulas and discrete groups. (Oslo 1987) Academic Press 1989 pp. 467-484

Titchmarsh, E. C , A divisor problem. Rend. Circ. Math. Palermo 54 (1930) pp. 414-429

Tsang, K. M., Remarks on the sieving limit of the Buchstab-Rosser sieve. Number theory, trace formulas and discrete groups. (Oslo 1987) Academic Press 1989 pp. 485-502

Weil, A., On some exponential sums. Proc. Nat. Acad. Sci. USA 34 (1948) pp. 204-207


III. REPRESENTATION OF AN EVEN NUMBER AS

THE SUM OF A PRIME AND AN ALMOST PRIME


On the Representation of an Even Number as the Sum of a Prime and an Almost Prime

A. Renyi

The problem concerning the representations of even number as the

sum of two primes and of odd number as the sum of three primes was

suggested in the correspondence between Euler and Goldbach in 1742.

Using his method on the estimation of trigonometrical sum,

academician I. M. Vinogradov [l] has proved the Goldbach theorem for

odd number in 1937. In 1938, N. G. Tchudakov [2] has proved, using

Vinogradov's method, that almost all even integers are the sums of

two primes. The approximate result of other type was obtained by

Viggo Brun [3] in 1920, who has proved, using the elementary method

of the Eratosthenes sieve, that every even number can be represented

as the sum of two almost primes, i.e., 2N = P, + P„, where P, and a) \ d \

P? have at most 9 ' prime factors.

A conditional result was proved by T. Estermann in 1932 [6],

namely every large even number is the sum of a prime and an almost

prime which has at most 6 prime factors. Estermann's result is

based on the assumption of the famous unproved Riemann hypothesis

for all of Dirichlet L-series. In this paper, I prove the following

theorem without any hypothesis.

Theorem 1. Every even number can be represented in the form

2N = p + P, where p is a prime and P an almost prime, i.e., P

has at most K prime factors, where K is an absolute constant.

The detailed proof will be appeared in another place, and we

give here only the main steps of the proof.

Riemann hypothesis can be avoided by the use of a new theorem

a 9 can be replaced by 4. See Tartakovskii [4] and Buchstab [5].

on the zeros of L-senes (Theorem 2) which is established by Ju. V.

Linnik's two methods; the method of the large sieve [7] and the

method contained in his paper [8].

In order to formulate Theorem 2, we introduce certain definitions.

It is known that every character belonging to the <j>(D) characters

mod D, where D is a square free number, can be represented

uniquely as the product of characters belonging to moduli which are

the prime factors of D. Thus if D = pq, where p is a prime and

(p,q) = l, then every character belonging to the modulus D is of

the form xD(n) = Xp(n) X q(n). where Xp(n) a"d Xq(n) are

characters belonging to moduli p and q respectively. If xD(n)

is not a principal character, then we shall call Xn(n) primitive

with respective to p. Clearly if Xn(n) is primitive with respect

to every prime factor of D, then it is primitive in the usual

sense.

Theorem 2. Let q be a square free number, A>c, ', k = og q + 1 ' log A

and kslogA. For all primes p such that A < p < 2 A and (p,q) = l, with the possible exception of not more than

A3/4

such primes, the

Dirichlet L-function modulo D=pq L(s, X) = I ^ ( s = a + it) ,

n=l ns

where x(n) 1S primitive with respect to p, has no zero in the

domain a > 1 — — , Itl < log D, where 5 > 0 is a constant. k+1

It follows, for example, from Theorem 2 that there exist

infinitely many primes p-| ,p2,... ,pn,... such that if x(n) is not

the principal character modulo p , then L(s, x) 0 for s = a + it,

0 > l _ i ) |t| < log p , where 6 > 0 is a constant.

c,,c9,... denote the positive absolute constants.

Let

H(2N) = I log p- expf-p^Si!!) ^ ( 1 )

p < 2N 2N (2N-p,B)=l

where B = n p and R is a given integer. Let

c 2 — L - - I |R (2N)| , (3)

log^N QeE w

where the set E is defined as follows: E contains square free

numbers of the form Q = p,p9.. .p„ (p, > p, > ... > p j , if [i/2]

c 2 0 for N>c 4, then Theorem 1

clearly follows with K=max(R+c2> c»). Hence the problem is

reduced to the estimation of the sum

I |R0(2N)| . (4)

We shall prove that

I |R (2N)| <-2L- . (5)

QeE g log3N

In order to estimate (4), we derive Theorem 3 from Theorem 2.

Theorem 3. Let q, be a square free number, A>c-| and

exp( (logx)2/5 ) < Aq] < /x .

Let k, = • + 1 , where p, is a prime, A<p .<2A and 1 log ( P ] /2 ) '

188

(p,, q.) = l. Suppose that k, < log A. For all primes p,, with 3/4 the exception of at most A such primes, we have

wwi) l X(p)l0gp-exp(-p^^)

p< x x < x (6)

for any character x(n) modulo D = p,q,, where x(n) is primitive with respect to p, and 6, > 0 is a constant.

It can be derived easily from Theorem 2 and the well-known J. E. Littlewood's formula (Cf. [8]) that

J[A(n) X(n) e_ n / Y = - j

2 + v » L , ij- (s, x) r(s)Ysds

2-ioo L (7)

From the well-known results of E. C. Titchmarsh [ 9 ] , A. Page

[10] , and C. L. Siegel [11] , i t fol lows that

P D ( X > = 7 7 ^ n + 0 (xexp ( - c 6 /T^gT) ) ({ u <j>(D) l og x °

holds for all D < exp(Cg/log x) with the possible exception of those values of D which are multiples of a certain integer D, which eventually may exist.

For the case D-./D, we have

pn(>0 = + o( xexp(-c c/logx)) D •(D)logx V 6 V

+ 0(—!— x e ' \*(D)

where E is any positive number and c depends on e only.

Furthermore, we need the Brun-Titchmarsh's formula (Cf. [9]):

(9)

D, is the modulus for which the corresponding L-series has the

Siegel's zero p = a+it in the domain a > 1 /log x

PD(x) = 0(-*—) uniformly on D < / T , (10)

Consider the sum (4). Let 2N=x and

SY(x) = I x(p)logp-exp(-pl5M.) . (11) x p<x *

2/5 If Q>exp((logx) ) and Q = p,q,, where p, is the

greatest prime divisor of Q and not an exception in the sense of

Theorem 3. Then from

P Q ( x ) - * f e ^ , ^ ) s x ( x ) ' (12)

we have

VX)=^VX) + 0 ( X )• (13)

This process may be continued for q- = PoP?» Q? = V ^ an<* so

on, until after a certain number of steps, s say, the condition in Theorem 3 is not fulfilled by q = Ps+-i qs+i • Then if

2/5 2/5 q <exp(logx) , we use (8) or (9), while if qs>exp(logx) ' and p , is an exceptional prime, we use (10). Hence the estimation of (4) is reduced to estimate the sums of the following four types:

I. xexp(-cfi/logx). II. - L - x ° <MD)

1 - (6,/k1+l) III. -*— . IV. x '

• (D) N The sum of terms with type I is obviously of 0( — ) .

log4N The sum of terms with type II does not exceed

* c

Nlog N • exp( -logD, - - ^ l o g N ) . (14)

Df Although the value of D-i is an unknown, but we can prove that the

i maximum of (14) for 1<D,<°° , N>Cy. and e = - does not exceed

- 1 - 4 8

N log3N

The sum of terms with type III may be calculated by noting that for any q, the number of exceptional primes p in any interval (A, 2A) does not exceed A3/4 and thus

I •~<:~^- • T>c (15) p*>T P _1 T 1 / 4

Finally, the following elementary property of E is needed for the estimation of the sum of terms of the type IV: the number of integers Q = pq, where p is the greatest prime divisor of Q, which belongs to E and satisfies p < q''* (k>l integer) does not exceed ( ^ / ( R h ^ ) .

Therefore we have proved that for sufficiently large R, the N sum of terms of any type does not exceed — for N>c p, i.e.,

log3N B

I IM2N)| < - ^ - , N > c8 . (16) QeE g log3N B

c N From (3) and (16), we have H(2N) > —%- for N>c-, and thus

log^N * Theorem 1 is proved. The number of solutions of 2N = p+P is not

c9N less than

log3N

It is natural to expect that the number of solutions will be of

The weakness of our result lies on that the sum '(-V) Vlog 2N / log'

E logp* exp(-p —9-^-) is used instead of Elogp in order that

the length of zero free rectangle in Theorem 2 can be decreased when the Littlewood's formula (7) is applied.

Similar to the proof of Theorem 1, we have

Theorem 4. There exist infinitely many primes p such that

p=p+2 is an almost prime, i.e., the number of prime factors of P

does not exceed an absolute constant.

Theorem 4 gives an approximation to the well-known hypothesis

that there exist infinitely many prime pairs.

References

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[HI

I. M. Vinogradov, C. R. Acad. URSS, 15 (1937) 291-294.

N. G. Tchudakov, Izv. Akad. Nauk SSSR, Ser. Math., 1 (1938) 25-40.

V. Brun, Skr. Norske Vid. Akad., Kristiania, I, 3 (1920).

V. A. Tartakovskii, Dokl. Akad. Nauk SSSR, 23 (1939) 126-129.

A. A. Buchstab, Dokl. Akad. Nauk SSSR, 29 (1940) 544-548.

T. Estermann, 0. Reine Angew. Math., 168 (1932) 106-116.

Ju. V. Linnik, Dokl. Akad. Nauk SSSR, 30 (1941) 292-294.

Ju. V. Linnik, Mat. Sbornik, 15 (1944) 3-12.

E. C. Titchmarsh, Rend. Cir. Mat. Palermo, 54 (1930) 414-429.

A. Page, Proc. London Math. S o c , 39 (1935) 116-141.

C. L. Siegel, Acta Arith., 1 (1936) 83-86.

(See Dokl. Akad. Nauk SSSR, 56 (1947) 455-458, also Izv. Akad. Nauk SSSR, Ser. Mat., 12 (1948) 57-78).

Translated by Wang Ivan

SCIENTIA SINICA Vol. XI, No. 8, 1962

MATHEMATICS

ON THE REPRESENTATION OF LARGE INTEGER A S

A SUM OF A PRIME A N D A N ALMOST PRIME*

W A N G YUAN (3E 5 t )

{Institute of Mathematics, Academia Sinica)

§1

In this paper, we shall give the detailed proofs of certain results obtained upon assuming the truth of the grand Riemann hypothesis. (Cf. [1] , [2] ) . First of all, let us state the grand Riemann hypothesis as follows:

(R) The real parts of all zeros of all Dirichlefs L — functions L(s,X) are < Ml.

From (/?) we derive the following131

(R*) Let (1,0 = 1. Then

ft* <P(4) f=/(mod It)

where lix = l . J 2 log'

Now we state the results as follows:

Theorem 1. Under the truth of (R*), every sufficiently large even integer is a sum of a prime and a product of at most 3 primes.

Theorem 2. Under the truth of (/?*), there exist infinitely many primes p such that p + 2% is a product of at most 3 primes, where \ is a given positive integer.

Theorem 3. Under the truth of (/?*), every sufficiently large odd integer N can be represented as N = p + IP, where p is a prime number and P is an almost prime of not more than 3 prime divisors.

* This paper has been published previously in Chinese in Acta Math. Sinica, Vol. X, No. 2,

pp. 168—181, 1960, but the Appendix is added during translation.

1034

Theorem 4. Let Zk(x) be the number of prime pairs of the form (p,p + 2l() not exceeding x. Then

pmp — 2p>2^ (p — iy/loffx \ log3* / p>i

Theorems 1,2,3. improve the results which were obtained independently and simultaneously by the author™ and A. H. BmiorpaflOB15'. Our original results were obtained by replacing- 3 by 4 in these Theorems.

It is well known that if n(x; \, I) is represented by

P ( * ; * , / ) = 2 « - t i ? i I o g ? , . pf

then Theorems 1, 2, 3 may be derived from the following weaker hypothesis ( JR**)

(R**) Let X be the character mod D. Then L(s,X) has no zeros in the domain

|*| <log1 D, a>— (s = o + it), 2

In this paper, p, p', p", • • •; pu p2, • • • denote primes.

§2

Lemma 1. If x>\ and z>l, then

(Cf. [ 4 ] ) .

Lemma 2 . Let / U ) = <P(*) TT tJZ^. If Kz<x, Ky<x, then p\k P ~ 1

s ^ = T n ^ 7 n ( i + T-i—)iogZ + ooogiog3,). (*.2>>-I P>1

Proof. Let <Kq)*= TT t=-^. Then p\t P ~~ 1

y fi®. = -y J W n (i + -±—\ - y1 ^ ^ y — - = U,2»)=l U.2y)=l tt.2y)=l

1035

= y J^(?) y /*20) _ K'/q

Jî 9 ( * ) 0 ( « ) L 2<?y <7 J ( « ,2 jO= l

2y «<« ?<K?) («,2y)=l

- - 2 & £ JJ ( \ + 1 W , 4- 0(loglog3*) = 2y ,«)- ^ p(p ~ 2 ) /

- | I I ^ TT ( l + , 1 „ , W + 0( loglog3x) .

Thus we have the Lemma.

§3

Let Ky^x be two given integers. Let

Go) a, q; a, ( l ,) (1 < i < r ) ,

where 2 < pi< — < p, < £ arc all primes not exceeding £ and not dividing 4.

Let P~(x, q, $) be the number of primes p satisfying the fol lowing conditions:

(2 ) ? < * , (i a «(mod ? ) , ? ^ o,(mod f,) (1 < 1 < r ) .

It follows from the Chinese Remainder Theorem that the system of congruences

y = a,(mod p,) (1 ^ i ^ r )

has a unique solution in the interval 1 < y < pi • • • pr. Denote this solution by a*. Hence P„(x, q, £ ) is equal to the number of primes satisfying the following conditions:

(3) p^x,p = a(mod q), ? ^ a*(mod ?,) ( K » ' ^ r ) .

1036

Theorem A. Let c > 0; P = fT pi • Then under the truth of

(12*), the estimation

p „ 0 , , , ? ) ), where / ( ^ ) = <p(^) TT • t\k P — 1

Proo/. Denote ^ (^ ) =<p(^)_1 . If (^, y ) = l and ^|/>, then it follows from (R*) that

7ct <p(?)?>(*) p=d(mod q) p=«*(mod A)

Let

/ W - O ^ r t * ) / ^ . K O ' *IP ( / .y)=l

(*.y>=i

where d\P. Then

p=«(modij) (SXmod () (W, y)=i

tf,.y)=i W,.y)=i |.=«>(mod J J ^ )

«i.y)=i W,.y)=i

o (xwiog* ( 2 U'l)2) = - ^ 0 + *•

Let

.KM*

^ ( 0 5 _ 5 KO / i f

U.y)=i

Then

(in, *)=1 (m, y)=(f», * )=1

(m, y)=l

For (</, y) = 1, we have

*l*l* •» *l*l» m«V* 1\"*K)

(*.y)=l (*.»)=! ml?

= J_ V MO y ,. ( r \ _ i . /*(<*)

rlf Cr. y)=l

Hence

W,.y)=l W,.y)=i

- S *(*>( 2 **<*))*-j. <llr <l* l f

( * .y)=i ( * .y )= i

By Merten's Theorem, we have

for rf|P and d<,$°. Hence

K = oO , /2log* • f^log2?).

Thus we have the Theorem.

§4

Let £ > 2. Let / < c < / + 1, where / is a positive integer. Then

v /W= y ^(")_ y y f2(») + y y / (") +

nil- (n, 2<iy)=l (», 2«y)=l pp'<^ ' ' * ' ' " (n, y) = l (>l» <»> 2 « y ' = l

1038

+ •••+(-!)' S S l/*Q)l -, « > . * • * / ( » )

= V / (a) _ y ! y iKn) ,

CP'-V". «»>-» C».2p'-^("«,)-l

xl/«

1°. If 3<«^6 and *1/"<tf <r0*v" then we take g = , ., and

log"* «• = < 1. It follows from Lemma 2 and (4) that

4

.<(. K») n<(c /(.») 2 , i w p — l , > a \ pip — 2 ) / • I f (•. ny) - i * > *

(• .»)- i

+ O(loglog3*) =-

« — 2 8« ,|W

T T P~ 7 TT ( l + - 7 - ^ — r ) l o g * + 0(loglog3*). M « ? - l » > i \ PKP — 2 ) /

Hence by Theorem A we have

< A(«) —cs£— + o (-?*?- • loglog*V 9(^) log 3 * ^9>(tf) log3* /

logu* (5 )

where 8» _r

« — 2 ( 6 )

A(«) e-

c„ = tf-r JJ IT ( 1 — -7 — 1, y denotes Euler's constant.

p>i

2°. If 6 < « < 1 3 and xUu<q<caxv', then we take £ = - ^ - and

log"*

«• = —-— < 3. Since

1039

v _J_ v jîil- y 1 y A») _

= o f y _!_ y ££»}) =

pin

»(s^lS-»(^ _ J „ f(py j ^ . , /(»)

therefore

V J - V ^ » ) _ y i y A^(») = 0 (i°%x\ £* * £;. M JZ. tV) ,i,t /GO ^ e >

From Lemma 2 and (4) we have

v p'C") > y J^") - y JL y ^") + o (lo«x\

n\r (« . 2«y>-l

" p\n P ~ *• P>* P\P — 2) P>1

Hence it follows from Theorem A that (5) holds for

(7) yi(«) = ^ — -er ( « < « < 1 3 ) .

2 4 4

Let £/ denote the root of the equation

— — 2 — —log = 0. 4 2 4

Then

</« " i < o , if I / > M > 3 .

Hence A(«) is decreasing in the interval (3, U) and increasing in the interval (£/, 13). By numerical calculations, we have

7.35 4 .

1°. If g = * 1 / " a n d 2 < « < 1 1

2 , then we take <r = y ( y - - ^ ) < 1

and £ = log3"*

Since

y j£i?l= y fKn)_m-2jr p-2 n / i \

+ O(loglogx),

therefore from Theorem A we have

P„(*, q, xu') ', \ (p - 1)V g>(*) log1 AT

(8) + o (*g"h?lBg*) = Mtt) y + o (*g7^1?*), \ log3* / g>(tf)losr* \ (p(a)log3* /

+

where

(9)

2 ° .

a — 2

If ^ = *1/" and

- < « < J___2_ 2 f

1

J__± 2 f

oo

( * > 8 ) ;

then we take £ = -—r~ and log3*

c = — ( ) . Since 2 \ 2 « /

— aTL" \ T _ ~u) T \ T " W o g 2 \ 2 «/J

log* + O(loglogx),

• I f (». y>=i

x TT n - ^ £

1041

therefore it follows from Theorem A that (8) also holds with

(10) vt,(«) = 2ver

•(W)-'-HWMi-i) Since

4 ( « ) = -7-A,(«) -du

— 16

(« - 2) < 0 (if 0 < c < l ) ,

- " 3 ( 1 - l ogr )

( 2 c — 1 — c l o g c > - < 0 (if 1 < c < 2 ) l2

for given 4 < f < 8, hence ^x(«) is a decreasing function for « ^ 1

2 v

§6

Theorem B. Under the truth of (/?*), the estimation'

P„(*, ?, *1/u) > 25.8096*-' IT ^ - ^ • p>2 p — 2 !pl«y

n (i 1—) 1 + o (-*&-) /<p(q)\og2x M o g 3 * ,

AoWi uniformly in (<«>), where q is a given integer.

Lemma 3 . Let r > r, > • • • > r„ > 1 £e « g/'v <?« set of integers. Then under the truth of (R*), the estimation

PX*,q,Pr)>^TJ{-\R\

holds uniformly in (w), where

E _ j _ y 1 + y> y 1 -<r 9 ( ? » ) « < ' 0<r, <P(P°)<P(PI>)

(/>„. y)=i

ssss-s- 1

.<f K ' l ' < ' , K ' 3 M<, <P(P<>)' " •<P(P««)

<*«P0-"V »)='

i? - 0 ( ( 1 + r ) ( l + r,)2- • • ( ! + r „ ) V / 2 J o g * ) .

1042

Proof. Let P„(q; pu •••,/>,) = P„(x,q, pr). Especially, we have P~(q) — »(*, q, a ) . The difference between P„(q; pu • • •, p,^) and ?-(<?'> Pu—ypr) is equal to the number of primes p satisfying the following conditions:

P < *, P = o(mod ? ) , p =£ a,(mod p/)(l < i < r — 1), p = a,(mod p f).

It follows from the Chinese Remainder Theorem that the system of congruences

fy = «,(mod p r ) ,

ly = o(mod q)

has a unique solution a* in the interval 1 < a* < qpT. If /v+y, then (a*, qpr)=*l; otherwise (a*,qpr) > 1. For the sake of brevity, we write all the («>f) as («n). Hence

P » ( ? ; pi, •'•, pr-i) — P-(«; pi, ••', pr) =

= f — P-(qPr', Pl, ••, Pr-l), Hpr\y, t < l , i f ? , | y ,

r

P»(q; pi, • • , pr~) — P » U ) — 2 P - C * ? - ; ft, • • •» ?—0 — 0r»

O < 0 < 1 .

Using this formula r times and with the restriction jB < rx, we have

P»(tf5 Pi, • • • , Pr) > P«(?) - S P-(*P-) +

Pa**

+ 2 2 P->(IP>PI>; pi,---, PP-I) - 00 + m ) , o < e «*p. y )= i

Let r>r{> • • • >rn>l be a given sequence of integers. Since

P„(.qp.- • -p*; pi, ••-, P»-i) < Pu.(qp.: • •?„),

therefore we have

P-Ctf; Pi, ••, P^ > P«(<7) - 2 p « ( ^ « ) + S S PoilP'Pld -a<r « < r ^<<•l

-2(1+1-

2 2 V" 2 *•-(**• ••p*)-U + 0(l + ri) ,-"(l + r.)1

«>0>—>M

1043

Since we assume the truth of (2?*), hence

* - ( « ; * , • • - , * ) > ^ M £ - 1 * 1 . <PKq)


The proof of Theorem B. Let s be a given sufficiently small

positive number. Let h — — + e. Then there exists 8a such that

( 2 —J-r < log(A + e ) < 0.452 = r ,

p+«y

for S>S0.

Let p, — pri be the greatest prime not exceeding x1'13. If 2 < ^ <

/ + 1 , then we denote the greatest prime not exceeding x13**-1 by p, , where pr is the least prime with the property that />^*1<*0

i*/'r,+1-Let n be an integer such that 2n>2t +r,+i. Let rk = r,+1(t + 1 < ^ < » ) . Then we have (cf. [4])

9>U)

where

E > (1 — 0.0073193) TT (1 7"V) >

> 25.8096*-'' TT ^ ^ TT ( l - , * J — + O f-fsS-V

K - 0(*2 u »* "*' log*) = 0(*2 u 13(*-'> log*) = o ( - ^ - ) .

Mog * /


§7

Theorem C„ Let a, /3 be two positive numbers satisfying 8 > 0 > 4 and $>*>!. Then

1044

*+«y

where q is a given positive integer.

Proof. Let n=[ logx] , «, = « + 2 ~ / ( 0 < / < « ) . Then

f>+«y x"'+l<j><*"' f>+«y

1 1

< 2 4 r ^ ) r ; c 7 * + oU*£bgbg.iogSiti). i _L Mog0/>/<pO)<p(.tf)log2* Mog3* «/ /

p+«y

Since /li ( « + O(-j j j = il1(«/) + 0\- J and Aj(«) is a decreasing

function, therefore

T, < A,(«,) g«f log^±l + O U&L log log*log*±i).

Since

"fj /tt(«/) log -*-> - f ^ <*« < S (A l(«,+1) - A,(«,)) max log

- ° G ) - < 0

"7+1

«7

hence

g T, < ( - ^ [ ' ^ ,„) * + O ( i ^ l o g log*). TTo \ q>(q)J" " /log2* Mog'* '

This proves the Theorem.

1045

T h e o r e m C2. Let 3 < « < ) 3 < 1 3 be two given numbers. Then

P j > , q, *«*) > P „ ( * . q, x ^ - —Cg-SMSI du + 9>(^) log2xJ- «

+ o ( - ^ l o g l o g A Mog * /

where q is a given positive integer.

Proof. It is evident tha t we may assume a < £ / < 0 . W e estimate the differences P„(x, q, x1") - P„(x, q, xuu) and />„(*, q, x1/u) -Pu(x, q, x1").

The difference between P„(x, q, p„) and P„(x, q, pm+i) is equal to the number of primes satisfying the fol lowing conditions:

p^x, p = <a(mod q), p =£ a,(mod ?,)(«' < m), p = a„+1(mod pm+i).

If pm+\\y, then by the definition we know that it is equal to P»(x)^Pm+upm)> otherwise it is equal to 0 or 1. Hence

?„(.*, q, pm) — Pu,(,x, q, p„+x)\ = P„(x , qpm+i, pm), if pm+ilr y;

Arrange the primes between xvu and xu° as follows:

p, < xuu < Pl+i < • • • < p, < xv' < p,+l.

Then

P„ (* , q, xuu) = P„<>, q, *«*) + 2 P"(*> fl*U 1, Pi) + O ( l ) .

Let » = [ l o g * ] . «„ = <* + » J ( 0 < ? M < » ) - And put

71

« - l

T - 2 P"(*> »'+>» ft) = 2 r -n+i*iy

Since pi<qpi+1<4qpi and A ( « ) is decreasing in the interval (a , [ / ) , hence

P„ (* , #»,+„ p , ) = 2 P„ (* , qpi+l, x l°" ) < l i I

< A(«„) y log-^±l + o ( ^ f log logxlog-^±>).

1046

Since

'log

therefore

S AGO log-**! - r ^ l du = o ( J L ) ,

Hence

p.(*,», *w) < PJ>, », *"•) + ( - ^ p * k l *,) -4- + \ ip(?)J« " / log 2 *

+ o ( - ^ - l o g l o g A

Similarly, we have

^ <Pk<lVu - /log2*

+ 0 ( f * f loglogA


§8

Let 4 < f < 8 , 2 < « < f be two given numbers. Let 9Jt denote the set of primes satisfying the following conditions:

P ^ * , p = a (mod q~), p ^ a, (mod pj)(» ^ t),

P & o,+/(mod p)+i)(j <:t — s),

where p,<x1/r<p,+u p,<x1/"<pl+1 and q is a given positive integer.

The number of elements of 9Jt is denoted by Mw(x, xv', x1").

Lemma 4. There exist sequences of integers («•>/) such that the number of elements of 3D! satisfying at least I congruences of

(12) p = a,+ ;(modp,+/)(K ; < < - * )

is at most

1047

Proof. Let Fy be the subset of 9K whose elements satisfy the congruence

p = a,+i(mod p,+i).

Now, we estimate the number of elements of Tj. If p,+i\y, then the number elements of F; is equal to 0 or 1. Assume p,+i\y. Denote the solution of the system of congruences

In = a,+;(mod p,+;) ,

l» = <*(mod q)

by 2,+;. Let

( « • » , ) 2,+;, qp,+i\ «,-(l < * < «).

Then the number of elements of T, is not more than P~t(x, qp,+,,x1/r).

If the element of 9H satisfies at least / congruences of (12), then it belongs to at least / different sets Tj. Hence the number of elements of 9H satisfying at least / congruences of (12) does not exceed

4 2 *-j(*» iP,+i, *1/ r).


Theorem D. The number of elements of 9Jt satisfying at most m congruences of (12) is not less than

Proof. Since

»<*

< s (-T-+0 " o(*,/") + ° (*1_i)'

therefore it follows from Lemma 4 and Theorem C t that the number of elements of 9)1 satisfying at most m congruences of (12) is not less than

M-(«, xl", *"") - — 1 — 2 P»/*> 19*1, *"') + 0(1) > m + 1 ;« - ,

1048


§9

It follows from Theorem B and Theorem C2 that

P.(x, q, *"<) > (25.8096 - P ^ A.) ^ +

+ o ( -^- log log*) > 8.4 g^* + O ( ^ l o g log*). Mog3* / 9 ( 3 ) log2* Mog3* /

( i ) Let x = y be an even integer. Let

(o>i) a = 1, # = 2; a, = *(i = 1, 2 , • • • ) .

From (9 ) we know that there exists a positive constant Xx such that

?„ , (* , 2 , *"«) - i - ( ( 6 M = 2 <fa\f*L + O (-£3f log log*) > 3 \J3 « /log2* Mog3* /

> (8.4 - 6.588) ^ - + O ( - ^ l o g log*) > - ^ - > 1 log2* Mog3* / log2*

for * > * ! . Hence it follows from Theorem D that for x>xt there exists a prime p such that \<p<x— 1 and * — /> has no prime divisors <,xut and has at most 2 prime divisors in the interval xm < p' < xm. Hence x — p is a product of at most 3 primes. Since x=*p+x — p , we obtain Theorem 1.

( i i ) Let x = y be an odd integer. Let

(«>j) a — * — 2 , ^ — 4; a, = *(»' — 1 , 2 , • • • ) .

From (9 ) we know that there exists a positive constant xt such that

> . £ i * 5 _ > 2 '2 log2* Mog3* / 2 1 ^ *

for * > * 2 . Hence it follows from Theorem D that for x>x2 there

exists a prime number p such that p<x — 3 and —-— has no prime

divisors <* 1 / < and has at most 2 prime divisors in the intervals xvs<

p' <* x1'3. Hence is a product of at most 3 primes. Thus We

have Theorem 3.

1049

(iii) Let \ be a given integer. Let

(o>3) a = l,q — 2; a / = — 2\ («'= 1, 2 , • • • ) .

It follows from (9) that there exists a positive constant x3 such that

for x>Xj. Hence from (9) we know that for x>x3 there exist not

less than -p*r~ prime numbers p in the interval Kp^x such that

/> + 2^ is a product of at most 3 primes. Thus we have Theorem 2.

From Lemma 2 and Theorem A with c = \, we have

pj,,2,^)<8n^n(i-7^-=U- + o(ri.iogkgA \ log3*/ «.IM? — 2 < > > , \ (*> — 1) Vlog** Mog3* /

Since

,1/. v * * V log3*/ K—— ——<r<*

log x lox jr

<»n ^ 4 n (i - T - S V ) ^ + ° (rr*0* lo**)' oi» ? — 2 i » i \ (p — l)vlog2* Mog3* /

we have Theorem 4.

REFERENCES

[ 1 ] Wang Yuan 1957 On Sieve Methods and Some of the Related Problems,

Science Record, Vol. I, No. 1, 9—11.

[ 2 ] Wang Yuan 1959 On Sieve Methods and Some of Their Applications,

Scientia Sinica, 8, 375—381. [ 3 ] HyaaKOB, H. I \ 1948 O KOHCffloft pa3H0CTH AJIH (pymcuHft W (*, \, / ) ,

HAS GGOP, cepua MameM., 12, 31—46.

[ 4 ] Wang Yuan 1956 On the representation of large integer as a sum of a

prime and a product of at most 4 primes, Acta Mathematica Sinica,

6 (4) , 565—582. [ 5 ] BHHorpaaoB, A. H. 1957 npHMeneHne f ( /) K peuieTy SpaTOOpeHa,

Mam. G6; 41, 49—80.

1050

A P P E N D I X

1°. We state the generalized weak Riemann hypothesis as follows:

(Ri) The real parts of all zeros of all DiricAlet's L-functions L{s, X) are <£ - 1 , where 1<<J<2.

Especially, (R3) is the well-\nown grand Riemann hypothesis.

For the sake of brevity, we denote the following proposition by (1, A):

Every sufficiently large even integer is a sum of a prime and an almost prime of at most A prime divisors.

Here, we state the refined result: 2 475

Theorem I. (1,3) may be derived from (Rtl), where 8{> 3 237

and (1, 4) is the consequence of (R>t), where 32> ' .

All the following results are obtained under the truth of (#»).

2°. Let 7 = - . Let x1/"^q^c<^x1/", where c0 is a given cons-o — 1

tant. Then the estimation

(1) P„(*, q, *"«) < A(„) / " * , + O ( ^ * l o g l o g * \

holds uniformly in («>), where

(2) A(u) = -%2-S (if , < « < 3 7 ) U-JJ

and

(3) A ( « ) = 25£l (if 3 7 < « < 7 9 ) .

og-

3°. Let v>2r/ be a given number and q = xv". Then

(4) P„(„, a, *'") < M«) <f + O / ^ l o g l o g ^

holds uniformly in («<>), where

(5) A1{u) = ^mêr u — j]

1051

for t) .iMi.i) for

• l . f

J- — J. <v<

(if f > 4 ? > ;

-1_± oo (if v < 4 J / ) .

4°. Let q be a given integer. Then

(7) P-(*, *, *'"•») > A(6.57) , y , + O (ga)t

where

(8) A(6.59) S* 2 7 X 6.453306.

The proof is similar to that of Theorem B with the essential difference that here we put t — 0.452 and A =15715 so as to obtain the more exact estimation

y . *«+»[(* + l ) f ] * " ^**+'[(* + O r ] * " , & ( 2 * + 4)! &> ( 2 * + 4)!

+ A'(8r)» V ( i * V X « " Y < 0.007183682. 18! w^ 4 6080/

5°. Let >/<a<0<6.57 be two given numbers. Let q denote a given positive integer. Then

p„(*,,, o > p.(«,,, «•") - y f ' ^ i A, +

(9) + o ( g £ l o g l o g * ) .

6°. Let u, v be two given numbers satisfying 2q < v < IO7 and ij<u<v. Let 9JI denote the set of primes p satisfying the following conditions:

p < *, p = a(mod q), p^. a;(mod p,)(» < x) ,

(10) p at+i(mod p2J+iXi < t — *)5

211

1052

where pt<,xu'<pt+u p,<x1/*<p,+1 and q is a given integer. Then the number of elements of 9H satisfying at most m congruences of

(11) p = a,+j(modpl+iXKi<t — s)

is not less than

(12) Pu(*, q, *"') - — J — ( ( ' AM ± ) y + o telog log*). /» + 1 \J» z /<p{q)log'x Mog3* /

7°. Computation of integrals.

(i) Al=rAoo,„=2<frp. </« M ~ 7 - i - » - 7 k > 8 » ~ 7

7 2)7 2ij

= 27<?r C'5 — 2Ver ( " / ( « » V ^

J4 , « / , « * > J4 W — 1 — log

2 2

< 2 7 t f r ( 2 /(4 + 0.02«) + 2 K5.46 + 0.02/)) M » O y-o '

< 2qer X 0.89050652.

(ii) Let v = 5q in 3°. Then

% - Jv'„a_j_)-i_4±_j.W4J--r) -\>7 « / 2 \ 7 u I 2\y u 1

= 207* r P * • = 209<fr P * ( « V * Ji (5 — 2*)(2sr — 1 — arlog * ) h

< 2 0 7 e r ( 2 K 1 + °-02') + S ^ 1 - 1 8 + 0.02;))

< 20>/cr X 0 .3972371.

8°. The proof of Theorem I. Let x=y be an even integer. Let

(ai) a = 1, q = 2; aj = x ( i = 1 , 2, • • • ) .

(i) Let 7 = 2.475. Then from 2°—7°, we know that there exists a constant *t such that

P„(x, 2, »«*) - -Iff" 4 iC5)*W. + 0 (£2dLlog , ) > 2Vjj5»_ « / log 2 * Vlog3* /

1053

£?«*_ + > P. <,, 2 , , - ) - ( r ^ s i * , + ± r ^ * ) _ \J» « 2 >"» » / log2*

+ O f - ^ - l o g log*) > 2?[(5.453306 — <rr(0.89050652 + Mog2* /

4- 1.9861855 + 0.738218)] - ^ £ + O f - ^ l o g log*) > log2* Mog3* /

> 0.05•££- + O f - ^ l o g log*) > 1, log2* Mog3* /

for x > Xi. Hence it fo l lows from 6° that for x> xt there exists a prime p such that p<x — \ and x — p has n o prime divisors ^x1/s*

and has at most one prime divisor in the interval x 1 / 3 ' < / > ' < * u* . Hence x — p is a product of at most 3 primes. Thus we have ( 1 , 3 } .

( i i ) Let 7 =-3.237. Then from 2 ° — 7 ° , we know that there exists a constant x2 such that

2\JJBL u /log2* W * / 5 » - l

> 0.01 - ^ £ + O ( - ^ l o g log*) > 1 log2* Mog** /

for * > * , . Hence it fo l lows from 6° that for x>x2 there exists a prime number /> such that p<x — l and x — /> has no prime divisor ^x1 7 5* and has at most one prime divisor in the interval *1/5» < p ' <

» T - 1

x "• . Hence a: — p is a product of at most 4 primes. Thus we have (1, 4) .

9°. It is well known that in the proof of Theorem I, the hypothesis (Ri) may be replaced by

(*.) £ ^(Z» Max «(*, D, /) - -MiL I - © (-?-), DZT/» « ~ V » "P(O) 1 viogr**/

where .<4 is any given positive constant and the constant implied by the symbol "O" depends only on 8 and A.

Similarly, (Rt) may also be replaced by

(« . ) £ / * D ) Max P(*, D, 0 - - 7 7 ^ O ( - — ) ,

1054

lot* t

where P(x, D, I) = 2 logp-e ' (cf. [1], [2]). p = 1 (mod D)

BapoaH13,41 first proved ( S u ) . Later, Pan Chin TongB1 obtained (Ris) independently, from which he deduced (1 ,5) . From Theorem I, it can be easily seen that (JRI.5) implies (1 ,4) . In other words, we have proved

Theorem II. Every sufficiently large even integer is a sum of a prime and a product of at most 4 primes.

Remar\. (1 , 4) has also been proved by Pan and BapoaH independently, but their proofs are more complicated than that given here, in fact, their proofs are based on (i?i.6) and (R^g) respectively (cf. [5 ] ) . I am grateful to Messrs. Pan and BapoaH for their kindly informing me of their results.

REFERENCES

[ 1 ] Pem>H, A. 1948 O npeAcraBjieHHH le-rawx «mceji B BHAC CVMMW npocToro H noqTH npocroro «mcna, HAH 6GGP, 2, 57—78.

t 2 ] Pan Chin Tong, 1962 On the representation of large even integer as a sum

of a prime and an almost prime, Acta Math. Sinica, Vol. 12, No . 1,

95—106 .

[ 3 ] BapoaH, M. B. 1961 ApHcpMeTHiecKne (pyHKuroi Ha peAKHX MHOHceer-Bax, ftoruadu AnadeMuu Hay* YaOGP, 8, 9—11.

[ 4 ] BapoaH, M. B. 1961 HoBbie npHMeHeHHH objibiuoro peuieTa KD. B. JlnHHHKa, Tpydu Hncmumyma Mamenamuxu, UM. B. H. Poxa-uoecKOto, Bbin. 22.

[ 5 ] JIHHHHK, lO. B. I960 AcHMirrOTHiecKaH (popMyjia B 3AAHTHBHOH npo<5-

jieMe TapAH JlHTTJibByaa, HAH 0C0P, TOM 24, J6 5, 629—706.

O n Representation of Even Number as the S u m of a Prime and an Almost Prime

Pan Cheng Dong

§1. Let N be a large even number and V(m) the number of

prime divisors of m. In 1948, Hungarian mathematician A. Renyi [l]

has proved that N = a+b, where V(a) = l and V(b)<K in which K

is an absolute constant. Under the assmuption of the Riemann hypo

thesis, Wang Yuan [2] has established that K<3. We shall prove,

in this paper, K< 5, namely

Theorem. Every large even number N can be represented as the

sum p+P, where p is a prime and P has at most 5 prime factors.

The proof of the theorem is based on the following

Fundamental Theorem. Let (SL, D) = 1 and

P,(N,D,l)= I log p . e-P(1 o9N / N> 1 p*N

pE i(mod D)

N + RD(N) . (1 .1 ) <J>(D) log N

Then

I |y(d) T(d) R.(N)| =(>(-==-) , (1.2) d < N V 3 - c d log5N

where e is any given positive number, x(d) the divisor function

and y(d) the Mobius function.

The proof of Fundamental theorem is based on the estimation for

the density of zeros of Dirichlet L-series.

We introduce the following notations: C-|,C2,... positive

absolute constants; e,e-|,e2>..- arbitrary small positive numbers;

B a bounded number but not always equal in different occurrences;

p,p,,p ,... odd prime numbers; XD(n) a character modulo D;

Xn(n) the principal character modulo D; p = 3 + U a zero u Xn Xn Xn of L-function L(s, x D)-

§2. Theorem 2.1. Let &>-. Let N(A, T, D) denote the

number of zeros of L(s, Xn) in tne rectangle A < a < l , |t|<D.

Then the inequality

N(A, T, D ) < C 1 5 D (2 + 4 C)( 1- A) T

3log 6DT

holds, where C is defined by

|L(|+it, XD)I < 3DC(|t|+l) , ( X D ^ X Q ) •

For the rectangle with length not too large compared with the

modulus of the character, the result is better than those of

Tatuzawa (Cf. [3]). We need the following lemmas.

Lemma 2.1. Let a and 6 satisfy 0 < a a except s = l. Further let | Re f (2 + it) | > m> 0 and

|f(a' + it')| <M Q t (a' >a, 1 < t' < t). Then if T is not the

ordinate of a zero of f(s), we have

|arg f(o+iT)| < 1 (log M T+2 + log 1) + ^

logQ " T 2 m 2

TT-B

for a>B (Cf. [4]).

Lemma 2.2. Let C < - + e,. Then 4 1

|L(l+it, X D)| < 3DC(|t|+l) , (xD ^ X J ) •

Proof. It is known that every character Xn(n) can be

represented as xD(n) = XD (") XD (")» where xD (n) is a primitive

character modulo D2, (D-|, D2) = 1 and D-jD,, < D1.

Let s = —+ it. Since 2

v Xp(n) _ X D ( n ) _ v Xp(d) "<d> \ ^

n > z n n > z n d D, d n d > t n b

( 2 . 1 ) and

XD (n )

I 2s nd> t n

< .

• oo

z I XD (n)

z/d < n z n

Set z = /D . Then

| L ( s , x D ) l s I XD(") XD(n)

n< z n

Lemma 2 . 3 . Let

n> z n < 3 ( | t | + l ) D

1/4 + e

( 2 . 4 )

„ y ( n ) x D ( n ) Py (s. z) = Py (s) = I £—

XD XD n < z n

where z > D l o g D. Then

I

Proof.

XD 2 < ( ^z + <()(D) l o g z

I P e + i t ) XD

V2

2 u(n) xD(n) u(m) Xpd")

XD n ^ z n^ + i t m < z mi~ j t

< *(D) I i L M + 2 ^ 0 ) J _ L ^ < <t»(D) log z + C,z m< n < z (nm)4 n<z n

n = m(mod D)

Lemma 2 .4 . Let 0 < 6 < 1 and

Then vs'z) = ys ) = L ( s ' x ° ) p x D

( s M

I | f ( l + 6 + i t ) | 2 < C 4 ( f 6 ~ 1 l o g 3 z + 6~ 2 l og 2 z ) XD

XD

Proof.

fv ( s ) = I X D (n )

XD n > z n s

where a = 1 y ( d ) . Therefore n d|n

d < z

o a Xn(n) a Xn(m)

I | f (1 + 6 + i t ) | 2 = I I " D I m ° ' XD

XD X D n > z n 1 + « + l t r ^ z m 1 * 6 " 1 *

217

a_a„ <0»(D) I - 2 ^ = 7 + 2*(D) I ^ 4 -

n> z n ^ + " z < m < n (nm) l +° n = m(mod D)

<*(D) I - ^ + 2 * ( D ) I lM_LM n > z n ' + f l z < m < n (nm)1 °

n E m(mod D)

* "KDjd1 + 1 2 ) ,

where

yl = y * (n) < 4 ' n > z n 2 + 2 6 "

I T 2 ( n ) u " 3 " 2 6 d u z z< n< u

1 , -1 ,«_3. < C„ z 6 log z ,

. 2 < 2 j T(n) x(m) < c D - 1 6 ~ 2 1 o g 2 z _

z < m < n (nm)1"1"6 J

n s m(mod D)

(2.5)

(2.6)

(2.7)

From (2.5), (2.6) and (2.7), we have

.-1 ,-1 ,__3. 2 ,„2_ I |f (l+6 + it)| < C4(Dz"' 6"1 logJz + 6~* log z)

*D *D

Lemma 2.5. Let G(s, z) = G(s) = n g (s), where g (s, z)

XD XD XD

2 = g (s) = 1 - f (s). Then G(s) has the following properties:

*D 1 1) G(s) is real for real s, and 2) Re G(2+ it) > j .

Proof. For the proof of 1), we may refer to [3]. Now we

proceed to prove 2). Since

an " D * " " I f ( 2 + 1 t ) <

XD I

n > z n 2 + i t

< V l M < 3 1 ° 9 Z

- L 2 " 7 n > z x\c <•

we have

Re G ( 2 + i t ) = Re n ( l - f j ( 2 + i t ) ) > 1 - ( n (1 + | f | 2 ) - l ) XD

XD \xn XD / D

10 D

*2 - < ! • £ > > f

Lemma 2.6. Let f , ( s ) , . . . , f (s) be an analyt ic and bounded

functions in the s t r ipe a < a < 6 - Let

F(s) = I | f n ( s ) r and M(o) = sup F(s) i = l Re s=a

Then

M(a) < M M ^ ' ^ M(6) ( a- a ) / ( 6- a) .

(Cf. [ 3 ] ) .

From the well-known Lit t lewood's theorem (Cf. [4]) and Lemma 2 . 1 ,

we have

N(A,T,D) < C5 6 -1 i if ( A - 6 + i t ) r d t

•TXD XD

+ max r I I f (s ) | a £ A - 6 Y „ XQ

| t | <T+2 D

(2.8)

219

In order to use Lemma 2.6, we introduce the function

h (s,z) = h (s) = ill cos"1 (j=)f (s) . XD XD S 2T XD

We have

Cfi|f (s)|e-(t/2T>< |h (s)| <C 7|f (s)|e-(t/2T) .

Let

H(s) = I |h (s)|2 and M(a) = sup H(s) . XD

XD Re s=a

Then from Lemmas 2.2 and 2.3, we have

H(l +it)<C 8e-dtl / T) I |fx (l +it)|

2

XD D

iCge-d*!/1) (|t|+l)2D2c(l |P)< (l+it)|

2 + C 8 D

C „ e " ( | t | / T ) ( | t | + 1 ) 2 D 2 c (z + D logz)

and therefore

M(l) = CgD2 cT2 (z + D logz) . (2.9) 2

By Lemma 2.4, we have

H(1 +6+ i t ) < C,0 e-WV I |f ( 1 + 6 + i t ) | 2

X D u

< C n ( 8~2 log2z + - 6"1 l og 3 z

Take 6 = — and z = DIogD. Then log DT

and

M(l +6) = C]2log4DT

M(l) = Cg D 1 + 2 c T2 logDT .

Set H(s) = F(s) , a = 1 and 6 = 1+6 in Lemma 2.6. Then we have

Ho) < M(l)<1+6-a)/(i+6) M O ^ 0 ' ^ ^

< ^ / ^ n i - o ) T4(l-a) l Q g6 D T

1 for — < a < 1+6. Therefore

M(A-6) < C H D ^ 4 ^ ^ ^ T4(]-A) log6DT

and by (2.8), we have

N(A,T,D) < C^McW-U T3 log6DT .

The theorem is proved.

(2.10)

Theorem 2.2. Let D< z (1/3)-E;

If L(s, xD) has no zero in

the domain

Then

1 1 5 — < a < 1 , |t| < log3D . log4/5D

, °° -fn/7\ -EoCogz) I I I XD(n) A(n) e

(n/z)| < C 1 ? 2 e 3

XD n=l

(RT)

1/5

where I' denotes a sum over all characters Xn(n) in which

L(s, xD) * 0 in (R^.

Proof. Since

I XD(n) A(n) e n=l

•(n/z) = - J _ 2-rri

r2+i~ , , ^ ( s , X n ) T ( s ) z s ds

2- i» L *D>

2iri J

i + i<*> i - (s, xD) r (s)zbds + I r ( P ) z

ioo L u p XD

*D

I r (p ) z + BlogD p XD XD

and

L 1 | r ( p x n) | z X ^ I . |r(P)|z« XD P.

*D 0 < e < l - [ ( C 1 6 ) / l o g 4 / b D ]

^^M^/ iogV^^ | T | < log3D

+ I | r ( B + i T ) | z 3

0 < 6 < l - [ ( C 1 6 ) / l o g 4 / 5 D ]

| T | > log3D

I | r ( 3 + i T ) | z e + z e ~ e 3 ( 1 ° 9 Z

i < 6 < l - [ ( C 1 6 ) / l o g 4 / 5 D ]

,1/5

< C|g log z

T| < log3D

2^

| < A < 1 - [ (C 1 6 ) / log 4 / b D]

1/5

I N(A, log3D, D)zA

+ z e e 3 ( logz)

S C l g log20z

+ z e

I §<A< 1 - [ (C 1 6 ) / log 4 / 5 Dj

- e 3 ( l o g z ) 1 / 5

,2+4c 1-A

we have

l' l Xn(n) A(n) e " ( n / z ) *D n=l

*D 

= P 2P 2"

p2 > p , s < 10 log log N ,

h-2 = ps-l qs-T qs-l = ps •

The numbers q,,q„,...,q , are called the "diagonal divisors" of

D. It is known that every character modulo D, where D is square

free, can be represented uniquely as a product of characters such

that their moduli are prime divisors of D. For example, if

D = p]q1, then xD(n) = X p (n) Xq (n). If X p (n) t x° (n), then

XD(n) is called to be primitive with respect to p,.

Theorem 3.1. (A. Renyi) [l]. Let q be a square free number,

A > C o n and put k = ° g q + 1. Suppose that k< log A. Then for £U log A

all primes p satisfying (p,q) = 1 and A < p < 2 A , with the

possible exception of at most A 3' 4 such primes, no L-series

formed with a character, which belongs to the modulus D= pq and is

primitive with respect to p, has a zero in the domain

C 1 |T ^ - < a < 1 , |t| < log3D .

log4/5D

We need the following lemmas.

Lemma 3.1. ^ |u(d)|T(d) <

C22 d< z <f)(d) log5z

V(d) > 10 log logz

Proof.

j | y ( d ) | T ( d ) ,, , - l O l o g l o g z ^ x 2 ( d ) . C22

d < z <j)(d) " d < z <f>(d) l og 5 z V ( d ) > l O l o g l o g z

Lemma 3.2. Let {p*} be a sequence of prime numbers with the property that there exist not more than A3/4 numbers belonging to {p*} in any interval (A, 2A). Then

p * i M p*-l 23

Lemma 3.3.

I X D ( p ) l o g P . e - ( P ^ N / N ) , 1/2 _ p > N U ^

Lemma 3.4.

I x D ( p ) l o g p e - ( P l o 9 N / N > p< N u

= \ x D ( n ) A ( n ) e - ( n l ° 9 N / N ) + BN1/2 . n=l

Lemma 3.5. For all D< exp(C25 /log x ) with the possible exception of those values of D which are multiples of a certain number D which eventually may exist, we have

v T -(plogN/N) I 1 og p • e v K 3 ' p< N

pE X,(mod D)

-c26/ioi~N + BNe db (3.1) <j>(D) logN

for (I, D) = 1. For the case D|D, the term

BN1-C(£)/De

<D(D)

should be added on the right hand side of (3.1), where C(e)

depends on e only.

Lemma 3.6.

P^N, D, i) < C2?N

*(D)

holds uniformly on D</Tf . 1 / 3 -e 2

Consider any D = p 1p 2 . . .p <N , where p, > p„ > . . . > p

and s<10 1oglogN. I f D > exp(log N)2 / / 5 , then s

J l and

p > DVV(D) > e x p ( l o g N ) V 3 ( 3 _ 2 )

q ] < p V ( D ) < p 1 0 1oglogN _

Therefore

logq, k1 = ~ +1 < 11 log logN . (3.3)

log — 2

When Theorem 3.1 is used to apply to (3.2) for fixed q,, we may

consider only the interval (A, 2A), where A = 2 l, k = 0,l,2

and 1 = exp(logN)1//3 . We call D > exp(log N ) 2 / 5 to be the

"condition 1". Further, we suppose that if p-, is the greatest

prime divisor of D and D=p.q., then p-, is not an exceptional

prime in the sense of Theorem 3.1 with respect to q,, which is

called to be the "condition 2". If these two conditions are all

satisfied, then from Theorems 2.2 and 3.1, and from Lemmas 3.3 and

3.4, we have

P i ( N ' M ) = *(J~)'pi(N,qrJl) + W) e x p (" e3 ( 1 o g N ) V 5 ) • <3-4)

If q, = p2q2 still satisfies the conditions 1 and 2, then

P (N,D,*) = ] P,(N,q?,Jl) + -^- exp(-e (log N)1/5) . (3.5) I *(P1P2) ' l *(D) s

I f for any integer m, the condit ion 1 is not s a t i s f i e d , i . e . ,

q <exp(logN) ' , then from Lemma 3.5, we have

225

P^N.D.a) N ^ BN , ,, ..J/5 , + exp(-e3(logN) )

<j)(D) logN <J)(D)

+ E,(q ) 1v ™' BN

1 -C(e)/De

where

w

<D(D)

1 . ^ D|qir

0 , if Dfq

(3.6)

If for a prime p +,, the condition 2 is not satisfied, i.e.,

p i is an exceptional prime with respect to q ,, then from

Lemma 3.6, we have

P^N, D, i.) BN *(D)

From Lemma 3.1, it yields

I 1 / 3 _ £ |p(d) T(d) Rd(N)|

d<N 2

(3.7)

1 1 / 3 -e ? d<N c

V(d)< 10 log logN

| p ( d ) T ( d ) R . ( N ) |

I ] / 3 _ e |y(d) T(d) Rd(N)|

d<N 2

V(d) > lOlog logN

< I 1 / 3 . e lu(d) x(d) R.(N)| + - ^ -d < N 7 2 l Q 9 5 N

V(d) < lOlog logN

(3.8)

From (3.6), (3.7) and Lemma 3.2, we have

I l / 3 - e 9 M d ) T ( d ) R d ( N ) | d< N l

V(d)<101oglogN

v / 3 M d ) | T ( d ) \ N e - ^ ° ^ ) 1 / 5

d < N 1 / 3 " £ 2 • « > I

+ x _ ( d l N l - C ( e ) / d e / l T (d) \

<f,(d) \ d<N <Md) /

+ N I 1 ^ 1 I —!— < - V » (3-9) d s N *<d> M o a N l 1 / 3 P * _ 1 l o g N

p* > e

and therefore the fundamental theorem follows.

References

[l]. A. Renyi, Izv. Akad. Nauk SSSR, Ser. Mat., 12 (1948) 57-78.

[2]. Wang Yuan, Acta Math. Sinica, 10 (1960) 168-181.

[3]. K. Prachar, Primzahlverteilung, Springer-Verlag, 1957.

[4]. E. C. Titchmarsh, The theory of the Riemann zeta function, Oxford Univ. Press (1951).

(See Acta Math. Sinica, 12 (1962) 95-106.)


Noted by the Editor: We omit the remaining part of the paper,

since from the fundamental theorem, it can derive that (1,4) by the

arguments of the Appendix in the preceding paper by Wang Yuan.

The "Density" of the Zeros of Dirichlet I-Series and the Problem of the Sum of Primes and "Near Primes"

M. B. Barban

Let ir(x, D, I) denote the number of primes in (1, x) which are = £(mod D). In this paper, a mean value theorem on ir(x, D, A) for (£, D) = l and "almost" all D < x 3 / 8" e is proved, where e is any pre-assigned positive number.

Theorem 1. For any given large number A, the inequality

lix I u (D) max - 3/8-e £(mod D) " U,D) = 1

TT(X, D, £) -+ (D) logAx

(1) holds.

By the combination of Theorem 1 and the Selberg's sieve method, we have

Theorem 2. Every sufficiently large even integer is the sum

of a prime and a product of at most 4 primes.

We sketch the history of this problem as follows.

In 1947, A. Renyi [1,2] proved the following theorem which gives

an important approach to the unsolved binary Goldbach problem:

There exists an absolute constant R such that e\/ery large

even integer is the sum of a prime and a number which has at most

R prime factors.

The constant R in Renyi's theorem depends on a lot of

constants contained in the analytic lemmas of Linnik [3]. It needs

complicated computation for evaluating the R, and its value will

be very large.

By the use of A. Renyi's method and some contemporary theorems

on the density" of the zeros of Dirichlet L-series, the author

proved in [4] that the conclusion of Theorem 1 holds if the sum is

over D<x '"~E, and it can derive already R = 9. Notice that

R = 3 has established by the assumption of grand Riemann hypothesis.

(Cf. [5]).

The further progress is connected with the refinement of the

"density" theorems of L-series.

Let N(a, T) denote the number of zeros of all the L-functions

mod D in the domain

a < a < 1, |t| < T , (2)

where the repeated zero is counted by its repetition.

The argument introduced in [4] shows that the relation (1) with 1 /a - F a sum of D<x ' can be derived by the estimation

c c N(a, T) « T ] Da(1 " a ) log 2 DT , (3)

where a, c-,, c? are absolute constants.

In spite of the solutions of Titchmarsh's divisor problem [6]

and Hardy-Littlewood's problem [7] by Ju. V. Linnik's new "disper

sion method", it is not without interest to notice that in the

famous conditional solutions of these problems [8,9], the Riemann

hypothesis may be changed by the "density" hypothesis, i.e., (3)

holds for a = 2.

In [4], we have used the "density" theorem of T. Tatuzawa [10]

which corresponds to (3) with a=6.

Out Theorem 1 is obtained based on a refined Tatuzawa's theorem

and also a deep theorem of Ju. V. Linnik [7] concerning the

estimation of sixth moment of the L-series on a half line.

We start from the following auxiliary lemma.

Lemma 1. Let 0 < a < 6 < 2 . Let f(s) be an analytic function

which is real if s is real, and regular in a > a besides s = l.

Further let |Re f (2+ it)| > m> 0 and

|f(a' + it')| < Ma (a' >a, 1 < t' < t) .

Then if T is not the ordinate of a zero of f(s), we have for, a>0,

| a r g f ( a + i T ) l ^ 7 l W ( l 0 g M a J + 2 + l 0 g i ) + Y • log -— v ' e-2-B

For the proof, we refer, for example, [11], 210-211.

We introduce the following notations:

Qz(s,x) = I M(n)X(n)n"s, f (s,X) = L(s,x) Qz(s,x) - 1 ,

n < z

hz(s,x) = 1 - fz(s,x) K (ff.T) = max £ |f (a+it, x) I |t|<T X

Z

All zeros of L(s, x) are also zeros of h (s, x)- Hence N(a, T) < N-,(a, T), where N-,(a, T) denotes the number of zeros of H (s) = nh (s,x) in the domain (2). Apply the well-known Littlewood's X z

theorem to this function. Since N,(a, T) is non-decreasing if a increases, we have

Nâ, T) < 6

,-1 rT

a-6 P - l Nâ, T) da < 6 ' Nâ .T) da

a-6

6 2TT

| l o g | H 7 ( a - 6 + i t ) | - log |H ( 2 + i t ) | } dt T I z I

-1 f2 - 1 . H (a + iT) - arg H (a - iT) da + 0(6 ) , 2TT J a - 6

z z

here the parameter 6 will be determined in the latter.

In the following, D is assumed to be sufficiently large and

z>D.

(4)

When a > 1 , we have

U s , X ) = I n - S I u(d) X (d) X(-5-) " 1 n=l din

d<z

= I X(n)n I u(d) = J X(n) an

n > l a J £ T < n ) ' n > z d | n n > z " n

d<z

where -r(n) denotes the number of div isors of n.

Since x(n) = o(n ) holds for any given e, we have

| f ( 2 + i t , X ) l z n > D

Now we apply Lemma 1 to (4) with

f ( s ) - H , ( s ) , a - (a - 26), •> ( a - 6 ) .

Since the D i r i ch le t series of H (s) has posi t ive coe f f i c ien ts ,

H (s) is real i f s is rea l . (See [10] , 301). Next,

Re H ( 2 + i t ) = Re n i l - f 2 ( 2+ i t , x ) } = 1 + Re n( l - f 2 ) - 1 I z x l z I X z I

> i - | n ( l - f 2 ) - i | > l - i n (l + | f 7 | 2 ) - l x z l x

> l - n ( i + D " 1 - 6 ) - l 1 > 2 I X I

[1 + D - 1 - 6 ) D > 1

Therefore m may be chosen to be —. Finally, 2

IH (s) | < n (l+|f (s, X)|2) < exp I |f (s, X ) | 2 •

X X

Therefore M + may be taken to be exp max K (a1, t) . a,t a < a ' < 2 z

We have still |Hz(2+it)| > Re Hz(2+it) > 1 and thus by the

combination of all the above estimations, we have the following

1emma.

Lemma 2 . I f z > D , t h e n

N(a , T) « 6~2T max K ( a , T + 2) . a - 3 6 < a < z

In order to estimate K (a, T) , we shall use the classical

method on the convexity theorem of analyt ic funct ion. We s ta r t from

the estimation of K ( 1 + 6 , T).

Lemma 3. I f z>D and 0 < 6 < 3 , then

K ( 1 + 6 , T) « 6~5 .

Proof. We have

I | f z ( l + 6 + i t , X ) | 2

X i t

I - ^ I y(d) I y(d)(-£-) i,D)=l (mn) l + 6 d|m d|n v n '

v L

m , n > z (m, n = m(mod D) d <z d <z

/ N b (n )

m>z mUS n = m(mod D) n 1 + 6

(m,D)=l n>z

where b (n) = l 1 . z d|n

d< n/z

Since (m,D) = 1, we have

I b z ( n ) . I I 1 = I I n< x n< x d|n d < x / z n< x

nsmfmod D) d<n /z nEm(modD) n E 0(mod d)

v . - nv - 1 . x x 1 og x « ) I -^- + 1 « - l o q x + - « a— hi-, \ Dd / D z d < x/z V Dd / D z D

and thus

K (1+6 T ) « ^ a i - y lM«I°LL ?2 ( l + 6 ) « M .

z ( ' z « 6 2 m ^ z m l + 6 z<5 62 z* 64

Since 6 log z « z , the lemma fo l lows.

l cn Lemma 4. Let z = D and max |L(-+it, x) I < MT for all

|t|*T 2

x(mod D). Then 2C

K z ( | , T) « M2 T ° *(D) log D .

Proof. It is evident that

K ( l T ) « M 2 T ° max £ ||Q (1+ it, x) |2 + 1

z*2 |t|<T X <2*z

2 2c0 « ITT u max *(D) I I E W - H M / J L ) 1 * Tl < T m,n<z (m,D)=l (mn)? v n '

n = m(mod D)

Since z=D, the lemma follows by converting the congruence to the equality.

Lemma 5. Let z = D and T>2. Then under the assumption of Lemma 4, we have

Kz(o, T) «

2cn 9 9 2(1-a) r , T u { N l V } logbD ( i < a < l ) ,

log D ( l < a < 4 )

Proof. By the convexity theorem of analytic function (Cf. [10], 309-310) and Lemmas 3 and 4, we have for ^ < a < 1+6 ,

i ? 2c0 ,(l+6-a)/(i+6) -5(a-j)/(J+6) K (a, T) « {M^T %(D)logD} 6

Let 6 = . Then the first part of the lemma follows by the TogD K

i/2 well-known estimation of L-series, namely M « D . The second part follows by Lemma 3.

1 c0 Basic Lemma. Suppose that max |L(-+it,x)| ^ MT for |t|<T 2

all x(moc' D). Then l + 2c„ „ 2(1-a) ?

N(a, T) « T u {ITD} log'D

1 3 If a > - + , the lemma follows by Lemmas 2 and 5.

2 logD 1 3

I f 0 < a < — + - — — , the lemma follows by the rough estimation N(a, T) « DTlogDT.

j.

In particular, set M = D4logD. The possibility of such choice

on M follows by the "approximate functional equation" of L-series,

cf., for example [12]. We have (3) with a = 3.

For "almost all" D, a considerably precise result is given by

the Linnik's estimation on the sixth moment of L-series, and it will

be used in the latter.

According to my previous work,, we know that Theorem 1 follows by

(3) with a = -| - e.

By Linnik's estimation on the sixth moment of L-series (Cf. [7]):

I I D1<D<D1 (1+ — L _ ) XD

I I |L(£+it, xD)l log20D

« D2(|t| +1) ° exp(log D] f

we have immediately

1-E

Lemma 6. Besides at most D, of D in the interval

D, < D< D, (1 + 1 — - ) , we have 1 ' log20D1

max |L(l+it, X)| « DV 6 + £ (|t|+l)

x(mod D) <•

c0

Hence if D is not an "exception" in the sense of Lemma 6, the o

Basic lemma gives the estimation (3) with a = -j+e. The result

given in [4] admits to consider only the case of D>x 1 / 7.

Let V denote a sum of D, where D denotes the "exception"

in the sense of Lemma 6. Then

I u2(D) max ir(x, D, £) - I l A -v 1 / 7 ^ n ^ v 3 / 8 - e *(mod D) *(D) X " " X (A,D) = 1

x V 7 < D - < x 3 / 8 - e " r

« x x X I n< log 2 2 x x 1 / 7 ( l + — l _ ) n < D < x 1 / 7 ( l + — I — ) n + 1

l og 2 0 x log20x

« x I 1

1 / 7 , ^ 1 " n< log 2 2 x X ( 1 +r1u" ) 3 log^x og

I' 1 « X

1/7 1 n 1/7 1 n+1 l o 9 A x

log20x 1og20x

Theorem 1 is proved.

Theorem 2 follows by Theorem 1 and Selberg's sieve method in the

forms of Wang Yuan [5] or B. V. Levin [16].

Theorem 3. The number of prime pairs, i.e., p and p+2 are

all primes, in the interval (2, N) does not exceed

(J6 + e) 2 n (1 -_" ) _J" ^ (5)

3 p > 2 (p-1)* log2N

where N> N(e).

The previous best record on this problem is due to A. Seiberg 1 fi

[13] with 8 instead of -=- in (5). It is not without interest to notice that Seiberg asserted that his result is the limit that may be attained by the "pure" sieve method.

Here we may use A. I . Vinogradov's argument [15] to derive that

either Theorem 3 holds with 4 - 2e instead of -y- or there exist

infinitely many prime numbers p such that p+2 has at most 2

prime factors.

Now we give the Selberg's sieve method by the form of the

following lemma.

Lemma 7. Let a,,...,a., be a set of integers such that

a = 0(mod d) f(d)

where f(d) is multiplicative and P = 0(1). If we use N to

denote the number of a 's which do not divide by any prime number

< z, then

N < z - l /(m)

m<z f l W

+ o

K Z f l

( log l o g z ) c I y 2 (d ) |R. | x(d) 2

where f-,(m) denotes the Mobius transform of f(m), and x(m)

the number of divisors of m.

The full exposition of Selberg's sieve method may refer to [10]

for example. In order to derive Theorem 3 from Lemma 7, we take

{a } to be the sequence {p-2}, where p runs over all prime

numbers <N, and z = N 3 / 1 6 " e .

The principal term can be evaluated by many well-known methods,

cf. [14], for example.

The remainder term is the sum

ir(n,d,z)

d d =

J £ N 3 /8

1 (mod

- e

2)

y2 (d) li N • (d)

T(d)

Theorem 1 implies that the part of the sum of d satisfying

A/2 N the inequality i(d) < log N, is « — — . And the remaini

part is logA/2N

I - ^ ( d ) T(d) d < N 3 / 8 " E "

x(d)> logA/2N

N I H ^ T ( d ) - I ^ d < N 3 / 8 - e d logA/2N log A / 2 _ 5N

Since A can be chosen arbitrarily large, the theorem follows.

Added in Proof. Theorem 1 was proved by the author in August 1961.

To estimate the Renyi's constant R with the aid of this theorem,

we refer to Wang Yuan's paper [5] which gives the known best result

on Selberg's "linear" sieve. It was published in Chinese and it

needs complicated computation in its applications. B. V. Levin

kindly informed me that the conclusion R < 4 can be derived by his

new work on Selberg's sieve and Theorem 1 [16,17]. And then Wang

Yuan confirmed that the same conclusion can be followed by Theorem

and his work [5], and the full exposition was published in an

Appendix of the English translation of [5]. It was noted that a

similar result to Theorem 1 was also proved by Pan Cheng Dong

independently, but it is not in the terminology of IT(X, D, I ) , but in a "weight" sum, from which the Theorem 2 can be derived, but

not the Theorem 3.

The author is grateful to Wang Yuan and B. V. Levin for their

systematically and kindly informing me of their results.

References

[1]. A. Renyi, Izv. Akad. Nauk SSSR, Ser. Mat., 12 (1948) 57-78.

[2]. A. Renyi, Dokl. Akad. Nauk SSSR, 56 (1947) 455-458.

[3]. Ju. V. Linnik, Mat. Sbornik, 57 (1944) 3-12.

[4]. M. B. Barban, Trudy Inst. Mat. Akad. Nauk UzSSR, 22 (1961) 1-20.

[5]. Wang Yuan, Acta Math. Sinica, 10 (1960) 168-181.

[6]. Ju. V. Linnik, Dokl. Akad. Nauk SSSR, 137 (1961) 1299-1302.

[7]. Ju. V. Linnik, Izv. Akad. Nauk SSSR, Ser. Math., 24 (1960) 629-706.

[8]. E. Titchmarsh, Rend. Cir. Mat. Palermo, 54 (1930) 414-429.

[9]. C. Hooley, Acta Math., 97 (1957) 189-210.

10]. K. Prachar, Primzahlverteilung, Springer Verlag (1957).

11]. E. Titchmarsh, The theory of the Riemann zeta function, Clarendon Press, Oxford (1951).

12]. Ju. V. Linnik, Mat. Sbornik, 53 (1961) 3-83.

13]. A. Selberg, Den 11-te Skan. Mat. Kong, (1949) 13-22.

14]. N. E. Klimov, Usp. Mat. Nauk, 3 (1958) 145-164.

15]. A. I. Vinogradov, Vest. Leningrad Univ., 7 (1959) 26-31.

16]. B. V. Levin, Dokl. Akad. Nauk UzSSR, 11 (1962) 7-9.

17]. B. V. Levin, Mat. Sbornik, 61 (1963) 389-407.

18]. Wang Yuan, Sci. Sinica, 11 (1962) 1033-1054.

(See Mat. Sbornik, 61 (1963) 418-425.)


New Results in the Investigation of the Goldbach-Euler Problem and the Problem of Prime Pairs

A. A. Buchstab

Goldbach-Euler problem concerning the representation of even

number as the sum of two primes has not been solved up to date.

With the aid of Eratosthenes sieve method and the theory of

Dirichlet L-series developed by Ju. V. Linnik and his successors,

one can prove that there exists an integer k such that every

large number 2N can be represented as 2N=p+n, where p is a

prime number and n has at most k prime factors. A. Renyi [l]

first established the existence of such k. k = 4 was obtained by

B. V. Levin, M. B. Barban, Wang Yuan and Pan Cheng Dong [2,3,4].

The corresponding results are also obtained on the problem of prime

pairs, i.e., there exist infinitely many prime numbers p such that

p+2 has at most k prime factors. In this paper, I shall prove

k = 3.

Theorem 1. There exists NQ such that every even number,

greater than N.., can be represented as the sum of a prime and a

number which has at most 3 prime factors.

Theorem 2. There exist infinitely many prime numbers p such

that p+2 is a product of at most k primes.

The proofs are based on the following theorem of M. B. Barban.

3 Theorem A. Let v be a number less than - and A be a given

8 3

positive constant. Then

I u (D) max n v a(mod D) U S X (a,D) = l

TT ( x , D ) - l i M . a 0(D) lnHx

where IT (x,D) denotes the number of primes sat is fy ing p<x and a

p = a(mod D ) , <j)(D) the Euler function, and u(D) the Mbbius function.

The proof of Theorem 2. Notice that it is well-known that the

Theorem 1 may be proved in a similar arguments.

Let q be an integer and 2 < p , < ...<p be prime numbers,

where p.|q and p < z < p r+T

Let a,a. ,a be a set of

integers such that (a,q) = 1 and p.fa., which is denoted by w.

We use P (x,q,z) to denote the number of prime numbers p satisfy

ing p E a ( m o d q ) , p£a,(modp.) ( l 0 and q < x ,

v 1/a

w«- <V

„v 1/a Pu[*«- (V

\(a) + o( (vln x - In q ) *

v 1/a rU(

x-"-{t)

TA( a) +o( ^ ) v (vln x - In q)s ' (vln x - In q)s

\> l/aN

+ r.(^> ^

c(q) 1i(x)

v 1 n x - 1 n q

c(q) 1i(x)

In x- In q

(1)

where B 0 is a constant which is independent on the choice of w,

7-T n *4 • d *(q) p|q P-2

P/2

A(a) > 0 if a> 10, c(q) =•

v 1/a r4 x , q' lT J < I u (D) max

Deft a (mod D) (a,D) = l

7Ta(x,D) 1i (x)

• (D) , (2)

v 1/a in which the domain fi = Q, I x,q, (-—) I consists of the numbers

x v

D such that D = qm, m < —-, and the greatest prime divisor of m

yv 1/a is less than (—)

q

We obtain the usual formulas for A(a) and A(a). The values

of A(a) with step size 0.01 when a<10, and A(a) = 9.999942 are

evaluated by the computer "Minsk-1" in the Moskow Pedagogical

Institute named Lenin. Two step functions AQ(a) and A„(a) are

defined by the so-obtained values, where AQ(a) = 0 if a< 10. It

is shown by Wang Yuan's work that one can prove the following

theorem by Buchstab's method.

Theorem C. Let B>1. If A(a) and A(a) are changed by

Ma) =

and

A(a)

max ( A(a), A(B)

Ma) ,

min ( A(a), A(B)

A(a) ,

6-1

a-1

A(z) dz J , if 1 < a<

if 0< a< 1 or a >

p"' Ml) a-l z

dz ) • if 1 < a <

if 0< a< 1 or a >

then the inequalities (1) still hold, where the remainder term also

satisfies (2) with the same fi.

Starting from AQ(a) and AQ(a), we have from (3) the follow

ing functions on the interval 0<a<10:

X0(a) < Aâ) < A2(a) < ... < A2(a) < Aâ) < AQ(a) .

By the successive iterations on the same computer, we obtain a

table with abundant values of certain functions A(a) and A(a),

where we omit the index for simplicity. In particular, we have the

following

a 3 3.1 3.2 3.3 3.4 3.5 A(a) 3.580161 3.58619 3.60711 3.64053 3.68437 3.73696

a

A(a)

a

A(o)

a

A(a)

3.6

3.79694

4.2

4.25994

4.8

4.81526

3.7

3.86318

4.3

4.34834

4.9

4.91197

3.8

3.93473

4.4

4.43877

5.0

5.00938

3.9

4.01079

4.5

4.53094

4.0

4.09072

4.6

4.62455

4.1

4.17392

4.7

4.71940

The fol lowing theorem may be proved by the s imi lar method used

for the proof of Theorem C.

Theorem D. Let — < a < 6 and v, <v . Then 8v 1

x 3 / 8 6 $ p < x 3 / 8 a p,, (x»P»P)

< !o . l i (x j ' M

In x '1

Aiildz + a-l z °(M-Mn

(4)

Theorem E. Let — < a < 3 < 6 and v , < v . Then

3/85 3/8a x < p< x

p (x,P,x3/86) <!° . IK*} In x

A ( - L ) ^ i a-l z+1

° ( 1 n 5 / 2 x ) (5)

Consider the intervals I = [ x n / 6 4 , x( 2 2 " n ) / 6 4 ] i f 4 < n < 1 0 ;

I n = [ x n / 6 4 ) X ( n + l ) / 6 4 ] i f "8<_ni20. a n d ^ = [ xn/64§ x (n + l ) /64 }

i f 4 < n < 1 0 . Suppose that c and d are corresponded to I - - rr n n n j i u 4 1 -r r i n (21 - n) .j.

and Lp , where c4 = - ^ ; cn = — i f 5 < n < 1 0 ; cn = n i f

18<n<20 ; and d = i 2 - L l l n l . Consider the function P(x,q,z) =

P ( x ,q , z ) , where a and a. are taken to be -2 .

Theorem F. Let ®(x) denote the number of primes p<x-2

such that 1) p+2 S 0(mod p.) for all p.<x 1 / 1 6; and 2) p+2

contains at least four distinct prime factors. The set of such p

is denoted by © . Let

n/64, S(x) = 1 c I P(x,p., x

4< n< 10 n p. G I 1

- - pi n

+ I c I P(x,Pi,x1/16)

18<n<20 n p.e I n ri n

+ I d I P(x,p.,p.) • 4< n< 10 n p.e L ^ 1

- - Ki n

(6)

Then © ( x ) < S(x).

Let M denote the prime numbers p < x-2 such that p+2 20(modp.)

for all p.<x1//16. For each p+2, where p e © ( © c M ) , it

A . u 0 ( ) <M (k3) (k4) can be represented in the form p+2 = p pg p p. m,

(ki) (k?) (h) (k/i) where p <P„ < p < Pr are four smallest distinct ra B y o

prime divisors of p+2 . Denote p ^ by x t / 6 4 < p ^ < x^t+1^64

if 4<;t<20; and x 2 1 / 6 4 < p^21 < x. Then S(x) = \ T(p),

where v

T(P) = 1 I c p. / (p+2) 4 < n < 1 0 n

1 p.e I , peM y-i n ' y n

+ I I c + d(p) , p./(p+2) 18<n<20 n

1 p.'e I K i n

peM means that p+2 20(mod p.) for a l l p. e x ; d (p )=d

V i ) (k,) i f p E L ( 4 < n < 1 0 ) , and d (p )=0 i f p * L

n for a l l such n.

For p e © , pick out the c and dn = d(p) in the sum T(p),

(kJ (k2) (k.) (k4) where p. is the prime divisor of p pfi p p» . We

obtain then its value U(p)<T(p). In order to prove the theorem,

it is sufficient to prove that U(p)>l for all p e ® . In fact,

S(x) > X T(p) > I U(p) > I 1 = ®(x) . pe © P e© p e©

In order to prove that U(p)>l for all p e ® , we consider

all the possible values of k,, k„, k3, k.. After 108 times of

evaluations on this function, we have U(p)> 1 for all cases.

The problem of optimum selection for the values of c and d n n

is essentially a problem of linear programming, i.e., to minimize

the linear form (6) under the condition U(p)> 1 . In (6), S(x) is represented as the sum of ten terms of the

form c I P(x, p., xs/ ) and seven terms of the form p. e I Ki n

d I P(x, p., p.). These sums can be evaluated by the Theorems H1 n

3 1 D and E, and the table on A(a). Take v. = -. We have x ' 8 107

(6)(x) < S(x) < 15.0607 B n — ^ - for x>x„. Denote by P(x) the uln^x u

number of primes p<x such that p+2 does not divide by any prime number < x / . Let a = a, = ... = a = -2 . Then

P(x) = P (x, 1, x1/16) > \ B x(6) - \ > 15.9979 B - \ -w 3 u lr/x

u l n 2 x

for x>x„. Let K(x) denote the number of primes p<x such that

p+2 is square free and has no prime factor < x1' '°. Then

K(x) < 0.0001 B Q — ^ - for x>x Q. ln'-x

Let F(x) be the number of primes p<x such that 1) p+2

has no prime factor < x 1 / 1 6; 2) p+2 is square free; and 3) p+2

has at most three prime factors. Then

F(x) > P(x) - (D(x) - K(x) - 2 > 0.937 B n - ^ — Uln2 x

for X > X Q . Since F(x) -»• °° when x •* °°, Theorem 2 is proved. In

order to prove Theorem 1, the p+2 in the definition of © ( x ) , M,

P(x), K(x), F(x) should be changed by 2N-p and we take a. = 2N

for p. (1 < i < r) in the definitions of P(x, p., xn'°^) and

P(x, p r p ^ .

References

[1]. A. Renyi, Izv. Akad. Nauk SSSR, Ser. Mat., 12 (1948) 57-78.

[2]. Wang Yuan, Sci. Sinica, 11 (1962) 1033-1054.

[3]. B. V. Levin, Dokl. Akad. Nauk UzSSR, 11 (1962) 7-9, Mat. Sbornik, 61 (1963) 389-407.

[4]. M. B. Barban, Mat. Sbornik, 61 (1963) 418-425.

(See Dokl. Akad. Nauk SSSR, 162 (1965) 735-738.)


The Density Hypothesis for Dirichlet L-Series

A. I. Vinogradov

§1. The basic aim of the paper is to prove the fol lowing

theorem.

Theorem 1. Let Nd(a, t ) be the number of zeros p of a l l the

D i r i ch le t L-series modulo d in the domain Rep> a, | Im pj < t .

Then besides at most n 1 - 0 - ^ integers in the interval D<d<2D,

we have -4

N d (o , t ) < ( t . l n D ) C ° ' e .Q2V+eW-°) , l < a < l , t > 1 ,

where E is any given small number.

This theorem is usually regarded as the density hypothesis for the average of Dirichlet L-series.

It yields from Theorem 1 and M. B. Barban's work [l] the following

Theorem 2. The mean asymptotic law on the distribution of prime numbers in arithmetic progressions:

I max 7T(x,d,£) — Li(x) *(d)

« x

(In x ) c

holds, where c is any given large constant and e > 0 any small

fixed number.

For many problems in number theory, Theorem 2 may be used instead of the grand Riemann hypothesis, in particular, it follows by Wang Yuan's work [3] or B. V. Levin's work [4] that

Theorem 3. Every large even integer m can be represented by

m=p+P.,, where p denotes a prime number and P, is an almost

ln -m

prime which contains at most 3 prime factors. More precisely, the

number of solutions of this equation is greater than Cn-<5(m)- m

where C„> 0 is an absolute positive constant and ©(m) the

singular series.

The similar results hold for the difference problems:

2k = p - P3 , k = l,2

Concerning the upper estimation for the binary problem, we have

the following

Theorem 4. Let m be an even integer. Then the number of

prime solutions p, q of the equation m = p + q does not exceed

(4 + e) • S(m) • —^—-, where e > 0 is any given small number and ln^m

S(m) the singular series.

There exist some well-known relations between the modulus of

L-series and the bound of its zeros. The number of zeros of L(s,x)

satisfies o(ln Dt) in a certain domain, and thus similar to

Theorem 2, by the argument of [l], we may prove a mean value theorem

for the power of divisor function x,(m), namely

I max I -zn(m) - A^(x,d)

d<xl-e ( M M m= 1(d) m< x

(lnx)c

n. where k and n are two given positive integers and A.(x,d)

denotes the expected principal term of the sum on T.(m).

This mean value theorem may be used to establish a generalized

formula of Linnik [6], namely

Theorem 5. The asymptotic relation

I x j o n + l ) • x(m) ~C U ) x l n k x m<x K K ' n

holds.

Notice that according to E. C. Titchmarsh's argument [11], it

gives easi ly from Theorem 2 a new proof of the Ju. V. L inn ik 's [7]

theorem which is the solut ion of Titchmarsh's d iv isor problem, i . e . ,

I T ( p - a ) ~ E(A) - x . p< x

The arguments of the proof of Theorem 1 are as follows. First, the main difficulty of the proof lies on the establishment of the following estimation, i.e., for any integer n > 2 and any Z in the interval D1/" < Z < D1/""1 , the inequality

2D 2n ? n

I I I Xd(m) < D^ Zn exp[(ln De)] (1) d = D x d^X 0 m < Z

holds.

It means that the mean sum for the values of a non-principal character does not exceed the square root of the length of the interval for summation.

The second step is to prove (1) for n > 2 . The method suggested in the present paper can be well used to treat the case of high moments of n> 4 , but not for the cases of n = 2,3. Notice that for n=3, the estimation (1) has been established by Ju. V. Linnik, and it will be useful for us essentially.

In a first glance, if we analyse the argument in §5 of the

present paper, we may get some strange phenomena.

For n = 2, (1) follows easily by the method given by §5.

References [1]. M. B. Barban, Mat. Sbornik, 61 (1963) 418-425. [21. A. I. Vinogradov, Dokl. Akad. Nauk SSSR, 158 (1964)

1014-1017. [3]. Wang Yuan, Acta Math. Sinica, 10 (1960) 168-181. [4]. B. V. Levin, Mat. Sbornik, 61 (1963) 389-407. [5]. Ju. V. Linnik, Izv. Akad. Nauk SSSR, 24 (1960) 629-706. [6]. Ju. V. Linnik, Abstract on Intern. Math. Conf.,

Edinburgh, (1958).

[7]. Ju. V. Linnik, The dispersion method in binary additive •problems, Leningrad Univ. Press (1961), Providence, R.I., (1963).

[8]. Ju. V. Linnik, Mat. Sbornik, 53 (1961) 3-38.

[9]. C. Hooley, Acta Math., 97 (1957) 189-210.

[10]. I. M. Vinogradov, Selected papers, Akad. Nauk SSSR Press (1952).

[11]. E. C. Titchmarsh, Rend. Circ. Mat. Palermo; 54 (1930) 414-429.

(See Izv. Akad. Nauk SSSR, Ser. Mat., 27 (1965) 903-934, ibid. 30 (1966) 719-720.)


Noted by the Editor: We omit the remaining part of the paper,

since the more elegant proofs of Theorem 2 may be found in the

following papers of Bombieri and Pan Cheng Dong in this book.

249

201

ON THE LARGE SIEVE

E. BOMBIERI

1. The purpose of this paper is to give a new and improved version of Linnik's large sieve, with some applications. The large sieve has its roots in the Hardy-Li ttlewood method, and in its most general form it may be considered as an inequality which relates a singular series arising from an

integral I | S{a.)\ida, where S(a) is any exponential sum, to the integral Jo

itself. Very recently Roth fl] made important progress on this problem by

proving the following results:

THEOEEM A. (Roth [1].) Let nt (1 <,j < Z) be distinct natural numbers not exceeding N, and let Z(N; q, a) denote the number of those «,• that are congruent to a (mod q). Let X^2, and let tPbea set of distinct primes p^X. Then, for any R^2,

S P S (Z(N;p, a)-ZlpY<ZN + ZXi l o g B + Z ^ & I R - * ,

where 10> \ denotes the number of elements of 0>.

In particular, if X^Nl(iogN)~l, then

ZpY. {Z(N;p,a)-ZlpY^ZXHogX. P*iX o - l x '

We shall improve a little upon this result by proving the following

THEOREM 1. With the notation of Theorem A, we have

S p | (Z(N ;p,a)- ZjpY^ 7max (N, X2) Z. v<X o-l v '

The general version of the large sieve which we shall consider here, and which contains Theorem 1 as a special case, will take the form of the following theorem, where now any reference to prime numbers and sequences of integers has disappeared:

THEOREM 2. Let the an be any complex numbers, and put

5 ( a ) - S a„e(na), (1.1)

where as usual e(i) = e*"u. Then we have

S £ | S(a/ff)|»« 7 max ( Z - y , X » ) S K l 2 - (1.2) «<JT o-i y<n<z

(a, B)-1 [MATHEMATUU 12 (1965), 201-226]

202 E . BOMBIERI

In order to deduce Theorem 1, take 7 = 0 and Z = N in Theorem 2, and take on = 1 if n = ni and an = 0 otherwise. Then 21 an |2 = Z (in the notation of Theorem A), and (1.2) gives

2 *2 | £ ( a / p ) | s « 7 max (N, X2) Z. PCX a - l

A simple computation shows that

PX\S(alp)\*=p £ (Z(N;p,ay-ZlpY, a-\ a=l v '

and Theorem 1 follows immediately. I t may be of interest to remark that Theorem 2 is not far from best

possible. Take first Z- 7 > Z 2 and o „ = l . Then |S(1) | 2 = (Z- 7)2 , and (1.2) gives the upper bound 1(Z- 7 ) 2 ; this shows that for Z- Y^X%

we cannot replace the factor 7 max (Z— Y, X2) by max (Z — Y, X2). Now take 7 = 0, Z=\, ax=\. Then |fl(a/g)| = 1, so that the left-hand side of (1.2) is

(S t(q))( 2 K l 2 } ~ - ^ 2 !«„ \q*ZX I \K<mSZ / W* y<n^z

2

This shows that for Z—Y<X2 we cannot replace the factor 7 by any number less than 3/772.

We may look upon (1.2) as an inequality involving additive characters, and we may ask whether there exists a similar inequality but with multiplicative characters instead. This is in fact the case, though the final result takes a different shape.

Let Q denote a finite set of positive integers, and put

M = M(Q)= max q, (1.3) qeQ

D = D(Q) = ma,xd(q), (1.4)

where d(q) denotes the number of divisors of q. Also, for any character v to the modulus qt let T(X) denote the Gaussian sum

T ( X ) = 2 x(«) «(«/«)• (1-5) o - l

rto|2=(

We have

(il{qlq*)q* if (q*,qlq*)=l, 0 otherwise,

where q* is the conductor of the character x (so that •% is the extension to the modulus q of a primitive character to the modulus q*). Note that |T (X) | 2 = 9 if x is a primitive character (modg), and |T(X 0 ) | 2 = /*2(<?) for the

principal character xo> a n d | T ( X ) | 2 < 9 always. The multiplicative analogue of Theorem 2 i s :

ON THE LARGE SIEVE 203

THEOREM 3. Let the an be any complex numbers, and let Q be any finite set of positive integers. Then

2 - r ^ r E | T ( x ) | 2 2 X(n)an\' see 9(9) x y<n«z I

< 7 D m a x ( Z - y , J W 2 ) 2 <*(»)K|2, t 1 - 7 ) y<n<z

where 2 denotes a summation over all characters x ' ° '^ e modulus q. x

In special cases it is possible to prove a slightly better result. In particular, if an = 1 if n is a prime and 0 otherwise, there is a result of the same general type but without D and d(n).

Theorem 3 has important consequences for the theory of the zeros of Dirichlet's i-functions and the distribution of prime numbers; in fact the large sieve was originally created by Linnik with a view to applications to classical problems in the theory of primes.

The principal result we shall prove on the distribution of the primes runs as follows. Let

^ ( z ; ? , o ) = 2 A(»), n < r

nâ (mod g)

where (o, q) = 1, and consider the error term in the prime number theorem for arithmetical progressions:

E(z; q, a) = </-(z; q, a)-zj9(q). (1.8)

Define E(z, q) and E*(z, q) by

E(z,q)=ma.x\E(z;q,a)\, (1.9) (o.a)-i

E*{z, q) = m&xE(y, q). (1-10) IKS*

THEOREM 4. For any positive constant A there exists a positive constant B such that if X sgz* (logz)_B then

2 E*(z,q)<z(logz)-*. (1.11)

We shall show that a possible value for B is 3.4 + 23. I t seems likely that this result could also be proved by appealing to

Roth's Theorem A; however, the use of Theorem 3 seems to be more appropriate.

We remark that nothing more precise than (1.11) can be proved even on the assumption of the generalized Riemann hypothesis, if we apply this in the form E* (z,q)<^ z* (log z)2 for q < z. One can say that Theorem 4 may serve as a good substitute for the generalized Riemann hypothesis in many additive problems involving primes. There are several instances

204 E . BOMBIERI

of this general principle, and Professor Davenport and the author have worked out in detail an application to the study of small differences between prime numbers in a paper submitted to Proc. Royal Soc. A.

Results like (1.11), such as

2 i*(q)E{z,q)<x{\ogzy-* (1.12)

for some positive constant rj have been claimed by various authors. The work of Linnik and Renyi [2, 3] on the large sieve led to an inequality slightly weaker than (1.12), for some 77. More recently, work on this subject has been published by BarbanfJ and Pan Cheng-Dong§||. However, Barban's work has been subjected to criticism by Pan Cheng-Dong§, and the present writer is unable to understand Pan Cheng-Dong's paperj|. (It appears that the exceptional set of primes in Lemma 1.2 of this paper depends on s and a, while the choices of a and s(=p) in (2.7) depend on D, with many possible choices for p, so that Lemma 1.2 is not applicable.)

Theorem 4 will be deduced from a new type of density theorem (Theorem 5 below) for the zeros of Z-functions; most of the known theorems in the so-called statistical theory of X-functions are contained in this density result.

Let N(a., T; x) denote the number of zeros of L(s, x) in the rectangle

a < a « l , \t\*kT, (1.13)

where £=SasS 1. Our principal density theorem is :

THEOREM 5. Let Q be a finite set of positive integers and let M and D be defined by (1.3) and (1.4). Then

2 •77-r2 |T(x) | i !^(a, T; X)<^DT(M* + jtf T)4U-«>/<3-2«) fog" (M + T)

(1.14)

uniformly with respect to Q, for | ^ a < l , T > 2 .

The proofs of Theorems 4 and 5 are self-contained, except for references to classical work of Landau, Littlewood and Titchmarsh, and we hope that they are given with an adequate amount of detail.

I t may be useful if we add some remarks on the significance of (1.14). I t was remarked by Littlewood tha t many results in the theory of Dirichlet's Iz-functions L(s, x), valid for fixed x a n d variable *, have

t M. B . Barban, Trudy Mat. Inst. Akad. Naulc Uz. S.S.R., 22 (1961), 1-20. % M. B . Barban, Mat. Sbornik (N.S.), 61 (103) (1963), 418-425. i Pan Cheng-Dong, Acta Math. Sinica, 14 (1964), 597-606 = Chinese Math., 5 (1964),

642-652. || Pan Cheng-Dong, Acta Math. Sinica, 13 (1963), 262-268 = Chinese Math., 4 (1963),

283-290.


analogues (" ^-analogues ") for fixed s and variable x (to variable modulus). A good example of this is

Em|-&(!+»«, x)i/loglog<>0, | - » C O

where the g-analogue is

hm£( l ,x) / log logg,>0 q-*oo

for the quadratic character v (mod q). I t is easily seen tha t our inequality (1.14) is related to the g-analogue

of the density hypothesis for the zeros of Z-functions. This hypothesis asserts tha t

2iV(a, T; xXqL+'T*1-**, (1.15) x

and its g-analogue is

'EN{a,T;x)<9"(1~")+"ri+t- (1.16) x

Here 2 denotes as usual a sum over all characters x (vaodq). The last x

two inequalities are unproved and probably very difficult. However, it is possible to prove something very nearly as good as (1.16). In fact, as we shall see later, it is possible to deduce from Theorem 5 the

COROLLARY. We have, uniformly for J < a < 1 and 2 < T < -\/X, that

2 ^,N(<x,T;x)<X1+w-'^Tl+c. (1.17) «^r x

Further,

Z P I ' . J ' l x X ^ ' ^ ' (1-18) <t^x x

uniformly for $ < a « l , 2^T^X*.

(1.17) shows that the density hypothesis (1.16) is true on the average with respect to q if 2 < T < (max g)*, and (1.18), shows tha t even more is true on the average with respect to q if J < a < 1 and T ^ (max q)2. The latter result is in a sense surprising, and the explanation is that Theorem 5 is really a new type of density result. We conjecture the following

Density Hypothesis. If \ < a < 1 and T > 2, then

2 2 * N(a, T; x) <X'4<1-a)+ ' T1+« (1.19) « t r x

uniformly in a, where 2 * denotes a sum over the primitive characters (mod q). x

Finally, the author wishes to express his deep gratitude to Professor Davenport for a discussion which originated this work and for his help in revising this paper.

206 E . BOMBIERI

2. In this section we shall prove Theorems 2 and 3. A rectangle in the (TO, n) plane of the type

y<m^z, y' <n^z'

will be denoted by R(y, z; y', z'), or briefly by R. Let c„, „ be a double sequence of complex numbers, defined when (m, n) is in the square Y <m^Z, Y <n^Z; to every such sequence we can associate a sub-rectangle

-Ro=-ô(ir» zo> Y>zo) of the square, depending on the sequence cm> „ and on Y, Z, with the property t ha t :

for every rectangle R = R(Y, z; 7 ,2 ' ) contained in R(Y, Z; Y,Z) we have

2 cm, n ^ 2 cm. i R R0

(2.1)

Plainly such a rectangle R0 always exists, though it need not be unique.

LEMMA 1. (Abel's inequality.) Let 6 m n be real numbers defined for Y <m^Z, Y<n^Z and satisfying the conditions

b ^ 0, bm „ — bm., „ ^ 0, 6 „ „ — &„, „ . i ^ 0. m, n — » m, n m+1, n ^ > m, n m, n+1 — » (i, I £ " *

I o m , n -Le< 5 = m a x 6 m n . 2"Ae»

°m+l, n — "m, n+1 + "m+1. n+1 ^ "•

X C„ „ b„ „ ^ -B XJ vm, n m, n ^ 2 c m , „ . (2.2)

Proo/. Put 6*, „ = &„,,„ if (w, »)e P 0 and &*_ B = 0 otherwise. Partial summation gives

ft

Hence

S c m > n 6 m . „ = S ( 2 CA.fc)(^ .»-^+i , B -6* > n + 1 +6* + 1 > B + 1 ) .

S c m , „ 6 m . „ U l m a x 2c m . „ )(2|&m.n-&m+i.n-*>m:.„+i + 6m+i,„+i|)

By (2.1) we have

max S c m , n U Scm .n . R I R < R„ I

and from the conditions (i) we have

2 I "m, n — "m+1, n — "m. n+1 + "m+1, n+11 R„

= 2 ("m, n ~ "m+1,n+1 — "m. n+1 + "m+1, n+l) R«

= ^ r + i . r + i ^ - S -

This proves (2.2).

255


LEMMA 2. Let c m B and R0 be as before, and suppose that -q > 0. Then

| 2 c m , n - ( 2 , , ) - i Z c \\Um-n)p)dp\*(^ -lMzCnJ. (2.3) ! « , Bo J~1 V / I \ X / l i t , I

where x = iirr)(Z—Y).

Proof. We have

W - ' S f W . f' c((m-w) i3)d JS= 2 ( - l ) * 7 S r T T i 2 c m . n ( ^ - » ) 2 * fl0 J -» V *=0 (^*+l ) ! i !o

where

» (-1)*(27TT))2* 2* l2k\

fc=l ( 2K+1) ! r=o \ r / J?0

The sequence bmn=(Z — m)r(Z — nYk~T satisfies the conditions (i) of Lemma 1, and Bti(Z-Y)*k. Hence, by (2.2),

\ênin(Z-mY(Z-nfk-'U(Z-Y)AY,cmX

Thus 00 (277TJ)2* » /2Jfc\ I I

*_i (^«-M)! r «o \ r I lflo I

/sinh a; \ I |

= hr-1 W-where x=inr)(Z—Y). This proves Lemma 2.

Proof of Theorem 2. Let Q

Sm,q= S «(sm/g) (2.4) o —1

be the well-known Ramanujan sum. Take

cm,n = amnn 2 Sm-n.q ( 2 - 5 ) (PUT

for (w. r i )e i J (y , Z; Y, Z). Choose T? SO that

x=im)(Z- y ) = min(l-3168, 2n(Z- y )X~ a ) . (2.6)

Then s inhx<2x , and (2.3) implies that

2*{Z- Y)\ 2* cm, n

'«o ' 2x — sinh a; |2<:m.„ [ ' e({m-n)p)dp\.

208 E . BOMBIERI

Since x

2x — sinha;> 1-4632

for 0 < x < 1-3168, we obtain

| S c m , J < m a x ( 7 ( Z - y ) , l - 4 7 ^ ) | 2 c m r n p e ( ( m - n ) i 3 ) ^ |

Let Wl„ n denote the interval | a — a/q\<r), and put

S{a; Y,Z)= £ ane{na), YKniZ

this being the same as S(a) in (1.1). We have

= 2 2 omo„ 2 2 f e ( ( m - n ) a \ d a y<in«Z 0 r<"=S20 ' « * * « - l JOT. . , '

(a, I > - 1

f «(a; = 2 2 « (a ; y , Z 0 ) S ( « ; 7 , Z0 ' )da. (o.«)=l

) This has absolute value

« J 2 2 f | | 5 ( « ; 7 ,Z n )P + | S ( a ; 7 , Z.')|«} At

< m a x 2 2 f |-S(«; 7,z)|2d<x « <t£X a = l J«D).., ( a . « ) - l

< m a x \\S(oi; Y,z)\*d«. i JO

= max 2 |«„ | a = 2 K l 2 -

Here we have used the fact that the intervals 2ftai, do not overlap, and this follows from the fact that q X, (a, q) = 1, and that the length 2TJ of 3J}„„ satisfies 2IJ ^ X - 2 by our choice of x in (2.6).

We have now proved that

| 2c m , „ U m a x ( 7 ( 3 - 7 ) , 1-47Z*) 2 K | » .

Since

2 cm.i R

2 c m , J


by (2.1), and

2 cm>n= S 2 2 2 am5ne(a(r»-»)/</) BSY.Z,

<a.a)-l

= 2 2 |-S(a/9;y,z)p( (o,9)=l

we obtain (1.2).

The proof of Theorem 3 is on similar lines, but we need also the following lemma relating multiplicative characters to Ramanujan sums.

LEMMA 3. We have

wAere tAe summation is over all characters Y to the modulus q.

Proof. If (mn, q)>l the result is trivial, since then x( w )x( w ) = 0 f ° r

every y. Now suppose that {mn, q)=l. Write 2 ' for summation over all residues (mod q) tha t are relatively prime to q. We have

2 | r ( x ) | 2 Y ( m ) x W = r 2 ' 2 x ( « ' « . ) v 0 n ) e ( ( a - 6 ) / < 7 ) X " h x 111'' v '

= Hq) 2 ' 2' e((a-6)/9) am=bn (mod 9)

= ^(g)S'e(A(»-w)/<?)

since all solutions of am = bn (mod q) are given by a = hn, b = hm. This proves Lemma 3.

Proof of Theorem 3. We take

<V„ = « r a 5 n E*-Sm_„,,, (2.8)

where 2 * means that we restrict ourselves to those q which satisfy (q, mn)= 1. Let

Sq(a; Y,Z) = S «„e(wa), (2.9)

< » , « > - l

fl»(a; Y, Z) = S «„e(na). (2.10) V<n<Z

d in

Choosing rj so that

x = i7rr)(Z- y) = min (l-3168, 2n(Z- 7) i t f"2) ,

210 E. BOMBIEBI

we obtain as before

\Zcm\<lm*x(Z-Y,M*)\-Zcmin f' eUm-n)p)dp\

= 7 max ( Z - 7 , J/2) S S ' f S,(a; r , Z 0 ) S > T T 7 Z 7 ) r f a | «cQ o JOT.., '

^ 7 m a x ( Z - y , ilf2)max( S S ' | \Sq(*; Y,z)\2da\. a \qeQ a JOT.., /

By a well-known identity, followed by the use of Cauchy's inequality, we have

|S , ( a ; y ,Z) |«« |zM(«*)S«>(a ; Y, Z)\' Idle I

«*te)E|<sw>(«; y>Z)!2 d i d

< - D S | S » ( « ; y.-Z)|2. (2.11) d=l

Hence

max( S 2 ' f |iS,(a; 7 , 2)j2da) ^D £ m a x P|,S«e(a; 7 , z)|2da

= D2max 2 W 2

d=l 2 y<n<z d in

= Z> 2 d(n)|an |2 . y<iusz

Theorem 3 follows easily, because by Lemma 3

jKy.z;y.z) j E Q ^ m a r<n<z (m.9)=.l (n,a)-l

= 2 2 2 am«n-7^rS|T(x)|2xWxW js«y<«szy<»«z <m; x

= S - T ^ - S l r f x ) ! 2 ! S x W o - f -««« ?(?) x iy<n<z i

3. In this section we give the proof of Theorem 4. We shall adhere to the following notation. For any character y (mod q) we shall denote by x* the (unique) primitive character associated with y and by q* its modulus, tha t is, the conductor of y. By £ * we shall denote a sum over the primitive


characters to the modulus q. The principal character (modg) will be denoted by x„. For any character % we define

0(*.X)- S x W A W . (3.1)

LEMMA 4. Let N be arbitrarily large but fixed, and let X0= (\ogz)N. Suppose that X < z*. For any D > 2 and any positive integer M, let QM denote the set of integers q satisfying

\<q^M, d{q)^D. (3.2) Then we have

S E*(z,q)^z(\ogz)-^ + zD-i(]ogz)3

ffS*

+ (logz)3 max J f -*E Tl*ma.x\if,(y,x)\, (3.3) X<,<MzLX QM x v*S*

for every (arbitrarily large) fixed A.

Proof. We have

ZE*(z,q)= E + 2 =2 i + S2> say. (3.4)

It is easy to estimate 22. Plainly

^2 ;<?,<*) < (logz) B S 1 < •(iog*)(i+*Mte)), n=a(mod q)

whence, by the definition of E*(z, q) in (1.10),

E*(z,q)<^z(logz)^(q) for 9^2.

Thus E 2 <tf*(z , l )+ £ 2(logz)M(9)

^ ( l o g z ^ + zflog^Z)-1 S %)#(? )

< 2 (log 2 ) - ^ + ZD-1 (log 2)3.

For the sum 2„ we express J?*(z, g) in terms of the values of </r(z, x)-We have

and this gives

*(q)B{z,q)*\f(z,Xo)-z\ + S W*,Jf)|. X>»Xo

Now

l0(2.Xo)-z|<«exp(-C(log2)»)<z(log2)-^-i

212 E. BOMBIERI

by the prime number theorem in its classical form; .and for x^Xo

</'(*. * )= S x*(m)A(i») mKz

Hence

= I/I(Z, Y * ) — 2 Y*(m)A(m) m^z

<m,«»l

= ^ ( Z > Y * ) + 0 ( 2 log?)

= +(z,x*)+o((\ogz)(\ogq)).

t(q)E*(z,q)<z(logz)-*-> + <{,(q)(]ogz)* + 2 max|./.(y, x *) | . X*Xo V**

I t follows that

X^zQogz)-*-* S l # ( j ) + *(log8)« + S - J — S m a x | ^ % , x * ) | •POT 7 c « x ? W ; x*x<> K* 2

<z( logz) -" + S -TJ-T- S max |^ (y , x *) | . ««CT PWJ X*X0 V=S*

Since g*|g we have d(q*)^d(q)^D, which implies that q*eQx, since q* > 1 for x ¥= Xo- Hence, collecting together all the terms containing the primitive characters x * belonging to the same modulus q*, we get

2 -TT-r S max |^ (y , x *) | = 2 2 * max|^(y, x ) | 2 W<7)-« £ Qx ? W ) X*XO V^Z «'*QX x v^z «€<jx

Now <^(g*r)»^(g*)<^(r)> q*<f>{r}(\ogX)~l, and X < 2 . Hence the last expression is

<(log*) 2 2 ( ?* ) - 1 S*max |^ (y , x ) | .

By the Siegel-Walfisz theorem (Prachar [4; ch. IV, Satz 7.2 and Satz 8.2]) we have

|*(*.x)K*e*p(-«(iog*)*) for X?tXo. a n (^ uniformly in q for <f< ( logzJ-^Z, , , where c = c(.ZV). I t is easily deduced that the same estimate holds for max|0(t/, X)|, with possibly a different value of c. Hence ***

2 (q*)-1 2 * max |^(y, X) |<s(log«)-l-«.

The remaining sum, that is, the sum over Xn<q^X, can be divided into <^ log X sums over intervals of the type 2m _ 1 < q < 2m. This gives

2 (9*)- 1 S*max!^ . (y , x )K(log Z ) max Jf"1 S 2 * m a x | ^ ( y , x ) | . j'eOx X l«3 Xt<MZX Q„ x V*iz r>Xt


I t follows that

Lj<<z(log2)-^ + (logz)3 max M~l 2 S * max |^(y , x% X*<M<X QM X V&

and on substitution in (3.4) we obtain (3.3). This proves Lemma 4.

Proof of Theorem 4. By a well-known explicit formula in the theory of primes (see «.(/., Prachar [4 ; ch. VII, Satz 4.6]), for X#Xo>

uniformly for q^z, 2 < T ^ z , where p=fi+iy runs through the zeros of L(s, x) with 0<j9< 1, multiple zeros being counted multiply. I t follows that

z? zlloz z)2

m a x | 0 ( y , x ) l < 2 —r- + ° + z> (3.5) v«* lyteT | P | J

uniformly for q < z', 2 < T < z', because | ^>(y, X) | <^ «' if y < «'• Consider first the contribution of any zeros with \p\ < \. The number

of these is <^logz (see Prachar [4; ch. VII, Satz 3.3]), and from the consideration of the corresponding zeros 1 — p of L(s, x) we deduce that

|p |>z-«

for any fixed positive e, provided q is sufficiently large (see Prachar [4 ; ch. VII, Satz 6.9 and Satz 8.1]). Hence the contribution of these zeros to the sum is

< 2 2*+C < (log Z) Z*+« < Z».

As regards the zeros with | p | > J , it suffices to take only those with 0 » * . We divide the range \y\<T into | y | < l and 2 m - 1 < | y | < 2 m for w t = l , 2 Then

lyKT P 2»-»«T lvK2»

Further,

S z^= 2 (** + z"logzrf<7) ly l<2" .

= z»JV(i,2»; y)+(logz) f ' ^ ( a , 2"1; x)z"rfa. /I

Using these results in (3.5), we obtain

max | if,(y, x)\ < z* + z(log z)2 71-1

+ (Iogz) 2 2-m(z*iV(i, 2m ; x ) + PiV(a, 2m ; x )z«da \ .

214 E. BOMBIERI

Hence

J f - 1 2 2 * m a x | . % , x ) | QM x vf

<JJf(z» + 2(log2)2T-1) + Jlf-1(Iog2) S 2-m(z> S S * t f ( i , 2 m ; x) 2—><T I QM x

J* e* x ;

<J(/(2» + 2(logz)22 r ' -1)+i/-1(log2) S 2 - m a x 2 2*JV(a,2'»;p ()j») V ' 2~-l«T a [Qj, x I

<JHf(2» + 2(log2)2T-1\

+ if-1(log2)* max (T ' ) -*max( £ S * ^ ( « , 2" ; x)2") • (3.6)

We now appeal to Theorem 5. Noting tha t for a primitive character X (modg} we have | T(X)|2 = ? > ^(<7), we deduce from Theorem 5 that

Z Z * t f ( « , 2 " ; X )< S ^ - 2 | T ( x ) | 2 ^ ( a ) y ' ; x)

< Z)T' (M3 + M 7")4<l-«)/(3-2a)(I0g Z)l.

Hence

max ( T ' ) - » m a x ( s S*-^(*» 2 " ; x)z"\

^ ( l o g z ^ m a x ^ + ifT)4 '1-"' ' '3-2*^*. a

The conditions we have imposed on M and T are M < X < z l and 21 < z*. The parameter Z) is at our disposal, and is independent of M.

Now take D = (log 2^+3, T = Jf(logz)^+5, A'<2*(log2)--4-6; the condition 21 < z* is satisfied since we retain the requirement that M < X. Substituting from the last inequality in (3.6), we obtain

i l / - 1 E 2 * m a x | ^ , x ) | QM X I/«*

< iJf2* + 2(log z)-A~3 + M-1 (log 2)^+2° max Jlf w—ws-w 2«. (3.7) a

Now

8 ( 1 - a ) ( 2 a - l ) 2

V ' - l = 2(l-a)- ^ ' =g2(I-a)-j(2a-I)2=j-2a2. 3-2a v ' 3 - 2 a

ON THE LABGE SIEVE 215

The function z" M3_2a" increases for a < a0 and decreases for a > a0, where a 0 = (logz)/(41ogitf). If M<z* then a 0 > 1; the function increases with a for \ ^ a < 1, and its maximum is

a i f - U 2 X 0 - * = a(logz)-»^.

If 2s < M < X( < 2J), the maximum of the function is

exP ( f logM + J(log2)2/(log-¥)).

Considered as a function of log M, the expression in brackets is convex, and its maximum for

£ log 2 < log M < Jog X is

max( j l og2 , j i ^ L + f l o g x ) .

We take X = z*(logz)_B, in accordance with the hypothesis of Theorem 4. Then the last expression is <logz — B loglog2+ 0(1), and it follows that

za j|/3-2«' ^ z ( ] 0 g Z\-\N + z ( ] o g 2 ) -B.

We take N = 2B, and suppose that B > A + 5 so that the earlier condition X < z* (log z)- ' 4 - 5 shall be satisfied.

Applying the results just proved in (3.7), we obtain

M-1 £ 2*max |^ (y , x )Ki f2» + z(logz)--4-3 + z(logz)^+2»-'B

for X0^M^X. Hence, by Lemma 4,

£ #*(z, g)<z(logz)-^ + z(logz)2^+23-B,

since D = (logz)^+3. Taking .8 = 3.4 + 23 (>^4 + 5), we obtain (1.11), and this proves Theorem 4.

4. In this section we prove Theorem 5, the proof being based on Theorem 3.

Let Q be a finite set of positive integers and let M and D be defined by (1.3) and (1.4); also let

z = M2, (4.1)

G(*,x)= S x W f W " - ' . (4.2)

/ ( s , x ) = i ( « , x ) G ( * , x ) - 1 - (4.3) We define

*»= n n (i-/2(*,x)K (4.4)

216 £. BOMBIERI

where j ^ \ e ( x ) = w | T ( x ) | 2 - (4-6)

Plainly F(s) is an integral function of s, since L(s, x) is an integral function for x^Xo> a n d e(x) is a positive integer. Also F(s) is real for real s, since then

/(«.*)=/(«. X). k(x)|2 = Nx)l2-

As we shall prove later, we have i^(s) ?£ 0 on a = 2, provided M is sufficiently large. We define arg.F(<7 + i<) in the usual way (cf. Prachar [4; p. 398]) by continuous variation front arg.F(2) = 0 along the line segments (2, 2+it, o + it), with the proviso that a,TgF(s) is to be increased by — rrm sgnt whenever the second segment passes through a zero of F(s) of multiplicity m.

We write

log+*={ £ log a; if x> 1, otherwise.

LEMMA 5. We have

jo. «€Q <p(g; ***,,

^ f {log|f(a+ftJl-loglÔS+ft)!}*

+ ^ { a r g f ^ + i r j - a r g i f ^ - f r ) } ^ . (4.6)

Proof. If /> is a zero of £(«, x) of multiplicity >n, then p is a zero of 1 — / 2 (« , x) and hence a zero of .^(s) of multiplicity a t least me(x)- Hence Lemma 5 follows from a familiar theorem of Littlewood (see Prachar [4], Anhang, Satz 8.1).

LEMMA 6. We have, for £^<x< 1,

f2 {&TgF{a+it)-a.TgF(a-it)}d<7<^M\+ f * " l o g + | f ( 2 + a + ( 2 - a ) e * )

Proof. If a > 1 it follows from (4.2) and (4.3) that

/ ( « . X ) - Zx(n)Az(n)n-',

where

A.{n)= S M W , \Ae(n)\tid(n).

dd.


Hence^ if M is large enough,

\f(2+it, x ) | 2 * : ( z d(n)n-*Y<(Z-»log2)2.

Thus \f(2+it, x)\ is small, and this justifies the earlier remark that F(2+it)^0.

We have

|log(l-/2(2+i/, x))|<z-2(logz)2; also i

2 -IM 2 IT(X)2|< S q<M* = z. Q*Q 9W) X*Xo Q*M

I t follows from (4.4) that

| l o g J ( 2 + tt)|< 2 - i j ^ S \r(Xf\\og(l-P(2+it,x))\

<Jtf!z- 1( log2) 2<Jl / ! . (4.7)

We follow the general line of argument of Titchmarsh [5 ; p . 180], For fixed t, let

g,{8) = l{F(8 + it) + F{8-U)}; (4.8)

then g,(s) is an integral function of*, and

g,(a) = 3iF(a+it)

by the reflection principle. Let n(r) denote the number of zeros of g,(sy, counted with multiplicity, in the circle | * — 21 sj r. For a ^ 2, we have

| argF(a+it) | « | a rg F(2+it)\ +(N+l)ir,

where N is the number of zeros of gt(s) on the segment a ^ s < 2. Thus, by (4.7),

larg F{a+it)\<^M\ + n(2-a).

By this inequality and Jensen's formula, if \ ^ a ^ 2 and gt(2) =£ 0, we have

| arg F(a + it)\do<^M\+ I n(r)dr

<^M}+ r~ln(r)dr

= M!+ i - J * l o g |f ,(2+ ( 2 - a ) e " ) | «W-log|fc(2)|.

Also, using the inequality

log+ (a + 6) < 2 + log+ o + log+ 6,

dO

218 E. BOMBIERI

we have

J l o g + U ( ( 2 + ( 2 - a ) e < » ) j ^ < 4 7 r + J l o g + I F(2 + it + ( 2 - a ) e « ) | dd

+ J*" log+ | F{2 - it + (2 - a) «-•») I

<l+\in log+ I F{2 +it + (2 - a) e*>) I dO,

since J(a—ti) is conjugate complex to F(a + it). Substituting in the previous inequality, we obtain the result of Lemma 6, provided — log|<7((2)|<^Jtf!. However, we can dispense with this hypothesis by considering also the function

h,{s)= — {F(s+it)-F(s-it)\.

Then \F(2+it)\i = \gl(2)\i + \hl(2)\^, and by (4.7) we have either -log\g,(2)\<^M\ or - log |A ( (2) |<a/ ! . The first alternative we have dealt with, and for the second alternative there is the same argument with h,{a) in place of <fr(*). This completes the proof of Lemma 6.

LEMMA 7. If x is a non-principal character (mod q), then

Hs,x)= S j f W i - ' + O l r ) (4.9)

uniformly for a ^ \, x Js 2q, 111 ^ xjq.

Proof. Write, as usual,

£(s.«0= 2 (n + w)-> n-0

for 0<w^l and <r> 1. By well-known methods (Titchmarsh [5 ; §4.11 or §4.14]) we derive the approximation

£(*,w)= S (« + « > ) - - - ! + 0(y-°), Ossnssi/ 1 — 8

valid uniformly for a > \, any positive integer y, and 111 < Try. We have „

i ( * , x ) = r ' S ^ ) { ( « , # a - l

and on substituting the above approximation we obtain

£(*.X)= S ^ ( n ) » - + 0 ( ?1 - ' r ) l

1

since 2 x( a) = 0 * ° r X^Xo- Taking x = qy + q, we have |*|<wy and we a - l

obtain (4.9) when x is an integral multiple of q. But we can omit the latter condition, since the sum over less than q terms can be absorbed in 0(qx-°).


LEMMA 8. We have

2 -j?TT S \T{X)\t\f(i+*.x)k(Xt + M\t\)\<n?[M + \t\). (4.10) 86 Q 9\1) X*Xo

Proof. By (4.3)

\f(s,x)\<l+\L(s,x)Q(s,x)\<l+l\L(s,xf + l\Q(s,xf.

Hence the sum on the left of (4.10) is ^ E J + E J + S J , where

2,= 2 -rj-r 2 |r(X)P, «eC PWJ X*Xo

S*= S -TJ-7 2 k(x)|2|£(H*Xx)|2. seg <PW1 X#XO

S ° = 2 -j^-r 2 I r(x)H « ( * + * , x)|*. «6 e <PW) x*Xo

Since | T(X)|2 < q we have Si < M*. Now consider 2 3 . We have

G(i+*.x)= S x(»)/*(»)»-,-a,

and on appealing to Theorem 3 with Y = 0, Z = z,an = /i(n)n_,_i/ we obtain

2 3 <Z> max(z, Jlf2) £ d(n)n-1<DJIf8( logJf) a .

There remains S2 . Here we first approximate to L{\ + it, x) by a finite sum, using Lemma 7 with x = Mi + M\t\, and q^M, which is permissible since then x > %q and 111 < z / j . We obtain

\m+it,xf<\ sxw»-H I n<x

2

+ 1.

Appealing to Theorem 3 again, with Y = 0, Z = a;, o„ = »-l~*', we get

9« Q 9 W ) X*Xo ' n « * '

<itf2 + I>max(a;, J/2) 2 d^n-1 ^D(M2 + M\t\)\og*(M + \t\).

Adding the estimates for E1( S2 , E3 we obtain (4.10).

LEMMA 9. We have, uniformly for a > l ,

2 - r j - r 2 |T(x)P|/(a + t* (x) |2<^log»(3f + |«|). t s « <P(9) XfXo

Proof. Let a; = Jkf* + J/111 as in the preceding proof. Then Lemma 7 gives

f(a + it, X ) = ( 2 x(") '»- , ,- a) ( 2 x(n)fi(n)n-"A - 1

+ 0(Mx-«\Q(o+it,x)\)

= 2 x(*)«»(*. «)»—* +O(jf1-"|0(a+»«.x)l).

220 E. BOMBIEKI

where an(x, z) = £/*(d) over d\n, nx^^d^z. Denoting by S the sum of the present lemma, we have

where

^= s -JLA S iTwi S x(»)«»*(«.«)»-^ ,r. « e g ? > ^ ; x*Xt> ' «<n««i I

see 9(1) x*Xo

By Theorem 3 applied to S2, with 7 = 0, Z = z,an=fi(n)n-a~a, we have

<S2<D max(z, JJP) 2 d{n)n-^^DM2.

Now consider /S^. We divide the range z < n < zx into <^ log a; intervals (2A_1z, 2''z), h=\, 2, ..., h0, together with a partial interval (2*oz, xz). By Cauchy's inequality,

2 x(wK(*.z)w-z<n^zx

2 »o+l 12

<(logz) 2 2 x(n)aH(x,z)n-'-it\ , A = l 2»- l « < » ^ 2 » « I

with the convention that the upper bound for n in the inner sum is xz when h = h0+l. This gives an inequality for Sl in terms of h0+l sums, the typical one being

2 -JL-2|T(X)H 2 x(»K(*,z)n-^|\

To each such sum we apply Theorem 3, with Y = 2h-1z, Z = 2hz (or xz), anân(x,z)n-'-il. Noting that \an(x, z)\^d(n), we obtain for the last sum the estimate

D max (2ft-1 z.Jtf2) 2 d(n)an2(x,z)n,-i<'<^D2>>z 2 d»(*)n-*'

2»- i2<n<2»j 2»-l *<n«2» s

(2*z)2-*»(loga:)7

<Z>z2-a '(loga;)',

since JV 2 <*>(«)<iV(log2V)7

n - l

and log(2*z)<loga;. On substituting in <S, we get

hc+l

S^Qogx) 2 Dz2-*"(\ogxy<^D(]oga;)».

Collecting the results, we obtain Lemma 9.


LEMMA 10. Let ft(s) /#(«) be regular functions of s in the strip a < a </S and continuous onto the boundary, and suppose that each of them -> 0 as | t| ->oo, uniformly in a. Let cx, ..., cxbe positive numbers, and define

J(°: A)-{£ jUl/â+i*)!**)*. (4.11)

Then J(a; Xu + tMv)^J(a; X)"J(fi; ft)", (4.12)

where w=03-a)/( |8-a), l>=(a-a)/03-a).

Proof. When K= 1 this is a theorem of R. M. Gabriel [6], and his proof extends without difficulty to the more general case. In the proof of Gabriel's Theorem 1, one considers

f K

2 ck<f>k(z)}k{z)dz JAB * - I

instead of $f>(z)f(z)dz, and follows the same line of argument but with an

additional application of Holder's inequality. This is the only modification of any substance that is needed.

Let us write for brevity

*(a,2> f 2 -£-r 2 WWII-P(*+it,x)\dt. (4.13) J-T q€Q f[q) x»*Xo

LEMMA 11. We have

0(a, T)-4DT(M2 + MT)«1-'**-*»lotf(M+T), (4.14)

uniformly for J < a < l , T > 2 . Also

<b(*,T)<^DT\og*{M+T) (4.16)

uniformly for a.2 1, T& 2.

Proof. For 7"> 4, put

/r(s,x)=/(*>x)/cos(«/n

JT(O; A) = (f°° 2 - 7 ^ 7 - 2 k(x)|»|/r(a+»U)r<&)\

For T > 4, £ « a « 1, we have iexpflil/THlcosfa/rêxpfltl/T), (4.16)

so that /r(«, x) i" a regular function of « for J < CT < 1 and -> 0 as 111 -> oo, uniformly in the strip J < a < 1.

222 E. BOMBIERI

From (4.16) and Lemma 8, we obtain

J T ( i ; l ) < f" e-'(„T S _J_ 2 \r(Xf\f{i + it,X)\dt

< f°° e-i'i'3*D(Jlf* + Jf \t$\og2(M + \t\)dt

<^DT(M2 + MT) \og2(M + T).

In the same way, Lemma 9 gives

U - » «e<2 0(9) x#Xo J

<{Z>T log9 (ilf+T)}».

On applying the two-variable convexity result of Lemma 10 with

« = £, 0 = 1 , A = l , /x = J, M = 2 ( l - a ) , » = 2 a - l ,

we obtain

JT(a; $-o)<^{DT(M2 + MT)\og*(M + T)f~*' {DT\og»iM+T)y-i

<^{DTlog»(M+T)y-°(M2 + MT)2-2", (4.17)

uniformly for £ < CT 1, T ^ 4 .

For every complex w and every A in \ < A < 1 we have

log+ | l -w 2 t< |« ; | 1 ' A . (4.18)

Hence it follows from the definition of (a, T) in (4.13) that, for \ < a ^ 1,

( D ( a , T ) < r 2 -rj-r- S | I - (X) | B | / (« ' + *«. X>|,|t,-^*»-J-T«eQ 0(g) **Xo

By (4.16) we can introduce a factor **•..

cos((<j + fO/ r ) | -l/8-<r)

into the integral, and this has the effect of replacing f(a+it,x) by

fr(o + it,x)- Hence

Now the first conclusion of Lemma 11, namely (4.14), follows from (4.17). In order to obtain the second conclusion, namely (4.15), we use again

the inequality (4.18), which implies that

<b[*.T)<\T S-r^-r 2 WixW^+it.xfdt


Now (4.15) for ct> 1 follows at once from Lemma 9. This completes the proof of Lemma 11.

Proof of Theorem 5. We write for brevity

jrâ'T)= S -JLET 2 \r(XfN(a,T; X). ««Q ?W x*Xi>

We start from Lemma 5 with j8=2; using Lemma 6 and (4.7) we obtain

2TTM\ y^rQ(a, T)da^ J {\og\ F(a+ it)\-log] F(2+it)\}dt

+ {arg F (<r+it) ~a.rgF (a-it)} da

< f \og+\ F(x + it)\dt

J-T + [2"log+\F(2 + iT+(2-*)ei0}\dd + MlT. (4.19>

The function Jf q(p, T) is a non-decreasing function of T for fixed a, so on integrating (4.19) with respect to T from 0 to 2T we get

/*2 /*2T f2 277Jlf!T ^rQ{a, T)do^2-nM\ ^VQW, U)dadU

< log+| F(oL+it)\dtdU

Jo J-u

+ P T (*"\og+\ F(2+iU + (2-*)ei0)\dedU

+ M\T\ (4.20) Obviously

f2T f t ; f2T log+l F(a+it)\dtdU^2T\ Iog+| ^ ( a + i ' O I * .

Jo J-U J-2T

Vr r ' l o g + l F(2+W+(2-a.)ei0^\dedV

<27rmax log+LF 2 + t t / + ( 2 - a ) e < f f rff/ e Jo I * ' I

T2T+2 < m a x log+|^(cT+iO|d<.

a o- 4 J -2

Using these results in (4.20), we find that

Ml \ jV~Q(o,T)do<^M\T+ max l o g + | / > + i7)|rf<. (4.21)

Also

224 E. BOMBIERI

We have

iog+i-F(*)i=iog+ n n i i - z ^ x H * * e« Q X+Xo

<m 2 -jTrt" 2 kWI'log+li-/»(*,x)l. »«Q 9W; x"xo

so that

\og+\F(o + it)\dt^M\<P(a,2T + 2) J-CT-2

by the definition (4.13) of <D(a, T). Hence, by (4.21) and Lemma 11, we have

rV c((T, T)do<^DT{Ma + MT)«1-*w-*'nog9{M+T), (4.22)

provided T. > 2 and J < a < 1. The function JVQ(O, T) is a non-increasing function of a for fixed T.

Hence if 0 < 8 < 1 we have

^ • g ( a + S, T ^ S - 1 f"+V< ,(a, T)da

^ S ^ - W ^ n - -W T ^ - ^ - ^ l o g V J f + T)

provided T > 2 and J < a sj 1. We take

S=l / log( .af+n and note that

4(l-a)/(3-2<x) = 4 ( l - a -8 ) / (3 -2a -28>+0(8) and that

(M*+MT)* = 0(\). Hence

JTQ(«, ^^DTiM' + MT^-'^-^Hog^M+T), (4.23)

uniformly for J + 8 < a < 1'. We have also

2 -7zrlT(x«)P^(«.y; *,)=#(«, r> s -^<T(iogr)( iog j f ) ,

(4.24)

where JV(<x, T) denotes as usual the number of zeros of f (a) in the rectangle | < «T < 1, 111 ^ T. On adding (4.23) and (4.24) we obtain the conclusion of Theorem 5, namely (1.14), for J + 8 4 a « 1.

Finally, suppose that J < a < ^ + 8. By a known result (Prachar [4; p. 223]), we have

S # ( « , T; x)<9(q)T\o%(M+T). x


Hence

2 -±-X\T(X)\*N(«,T; X)< S qT\og(M + T)<M*Tlog(M + T). QeQ <P\<1) X I^Q

This inequality is superior to (1.14) when £ ^ a ^ £ + S, in view of the definition of 8, and the proof of Theorem 5 is now complete.

We now deduce the corollary (1.17) and (1.18). Each character x mod q arises from.a primitive character \* modq*, where q*\q, and

N[*,T; X) = N(*,T; **) .

Since the number of values of q for a given q* is at most Xjq*, we have

2 ZN(«,T; X )< 2 Xlq*I,N(*,T; v*). 9«AT x Q'^X X'

Dividing the sum over q* into intervals (2A, 2A+1), we get the estimate

( logX)maxXJI/-1 £ ^.N^T; x*)<ma.xXM-1T(M2 + MT)P (XT)', M^X 8*«Af x* MUX

wherej9 = 4 ( l - a ) / ( 3 - 2 a ) . Thisis

<max(Z1+'2n+^+*, Z^+«T»+').

Since T < X* we have

Also 2/3^ 1 + 2 ( 1 - a ) . Hence (1.17). For (1.18) we note that £«;£ when | ^ a < 1.

We end the paper with two simple remarks about Theorem 5. I t is possible to prove many other similar inequalities, for example

°*Q X

which is better than our Theorem 5 if a > 7/10 and T is not too large. The inequality of Theorem 5 is analogous to another result of Ingham (see Titchmarsh [5 ; Theorem 9.19(B)]), and, by a happy circumstance, it gives a useful bound in the whole range T < M1+t; this seems to be essential in the proof of Theorem 4.

References

1. K. F. Roth, " On the large sievo of Linnik and Renyi ", Mothematika, 12 (1965), 1-9. 2 . Yu. V. Linnik, "The large sieve", Doklady Akad. Nauk SSSR, 30 (1941), 292-294

(in Russian). 3 . A. Renyi, " On the representation of an even numbor as the sum of a single prime

and an-almost prime number", Izv. Akad. Nauk SSSR, Ser. Mat. 12 (1948), 57-78 (in RuBaian); also American Math. Soc. Translations (2), 19 (1961), 299-321.

4 . K. Prachar, Primzahlverteilung (Springor, 1957). 5 . E. C. Titchmarsh, The theory of the Riemann zeta-function (Oxford, 1951). 6 . R. M. Gabriel, " Some results concerning the integrals of moduli of regular functions

along certain curves ", Journal London Math. Soc., 2 (1927), 112-117.

Istituto Matematico, Via C. Saldini 50, Milan.

(Received on the 21th of May, 1965.)

274

Noted by the editor: It follows (1,3) from Theorem 4 and the

arguments of the Appendix in the preceding paper of Wang Yuan.

Vol. XVI, No. 2 SCIENTIA S IN ICA May, 1973

ON THE REPRESENTATION OF A LARGER EVEN INTEGER AS THE SUM OF A PRIME AND THE

PRODUCT OF AT MOST TWO PRIMES

CHEN JING-RUN ( S S S )

(Institute of Mathematics, Academia Sinica)

Eeceived March 13, 1973.

ABSTRACT

In this paper we shall prove that every sufficiently large even integer is a sum of a prime and a product of at most 2 primes. The method used is simple •without any complicated numerical calculations.

I. INTRODUCTION

For brevity, we denote the following proposition by (1 , o ) : Every sufficiently large even integer is the sum of a prime and the product of at

most two primes. The sieve method and some results in the distribution of prime numbers have been

employed to prove the above proposition by many mathematicians, for example, (1 , c) Renyi1", (1 , 5) Pan Cheng-tung™, Bap6aH[3), ( 1 , 4) "Wang Yuan141, Pan Cheng-tung,5], BapoaH1'1,

(1 , 3) ByxmTa6[71, A. H. BHHorpaflOB181, Bombieri151. In a previous paper of the author1101 a sketch proof of ( 1 , 2) has been given. In

this paper we shall give a detailed proof of ( 1 , 2) .

Let x be a large even integer, h be any even integer, and Cx be J J - J J •

1 \ p>1

(P - DV-Let P r ( l , 2) be the number of primes p satisfying the following conditions:

x — p = Pi or x — p = p2p„

where p,, p2, p3 are primes. Let xh(l, 2) be the number of primes p < x satisfying the following conditions:

p + h = pi or p + /t = piPi.

In this paper, we shall give a detailed proof of

P , ( l , 2 ) > 0.67xCx ^ N Qogxy

In [10] we proved ( * ) with factor 0.098 instead of 0.67. By the same method we can prove that

158 SCIENTIA SINICA Vol. XVI

x 4 ( l , 2 ) ^ - 0 - 6 7 ^ (log x)2

The method used is simple without any complicated numerical calculations. In proving Lemma 9, a result from Richert1"1 and a well Icnown Bombieri's theorem are employed.

Now we state our results as follows: Theorem I. Every sufficiently large even integer is a sum of a prime and a

product of at most 2 primes, and it is found

0.67xC, P*(l, 2) >

(log x)2

Theorem II. There exist infinitely many primes p such that p + h is a product of at most 2 primes, h being any even integer, and

(log xy

The author is indebted to Professor L. K. Hua for his encouragement and to Professor S. H. Min and Professor Wang Yuan for their great help.

II. SOME LEMMAS

Lemma 1. For any real number x > 1 we denote by [logx] the integral part of log x. Set y > 0, and

4>(y) = _i_ r - £*« .

V (logx)1 1/

Set logx > 10J, and y > ^«°*'>-"•'. Then it follows

1 - x"0-1 < <P(y) < 1,

and $(j/) is non-decreasing for all y ^ 0. Proof. Besides the evident relation <P(,y) = 0 for 0 ^ y =S 1 we need

JL ( t f - (^){(log yY + ± ( - l y r - C r - i + D O o K ^ l ( 1 )

We now proceed by induction with respect to r. It is obvious that -*— (— ) =

(—jjlogj/ > and (1) is therefore true for r = 1. Suppose (1) has been proved

for r = 1, • • •, 8. We have

J?L ^ = _£. L-f-Oog^)! . v (-l) ' 'S---(S-'- + l)(logy)*--\]

_ « - f Qogy)"1 + v ( - D ' g - - - ( ^ - t + i)(iog?/)w- ; dogj/)^

No. 2 CHEN: PROOF OF (1 , 2) 159

+ 2 ( - 1 ) ; + ' S - • -(g - i + l ) ( i + l)( l0gj/)S- l _ ( £ \ {( logj , )^

_ (s+i)(iogi/)s, (-D^(s+i)i , y. K-iys-••(s-i+ixiogyy+i-!

, (-iya---(g + 2 - o CO'

s + 1

*fl»J^}-(£){(]**)•

,. V* (-Q'OSf + D- • <8 + 1 - i + l)(logy) s+ ' -- 1 _ i = 1 to' '

Consequently, (1) is also true for 8 + 1. We have

V 3 / l [log a:] I J lata""*1 \coJiv~-Uos,?-'

1 p— (log x) ' (logy) ^ X

l[loga;]!JJo

for. t / ^ 1 , which, by using also the relation $(t/) = 0 for 0 < y < 1, puts the non-decreasing property into evidence. From logs > 103, and el°'ll+x} =£ e*i<*2

w e obtain

0 < 1 — <P(v) = { }(" e-'X1""*1^ l[loga;] \ihiogxy\iogy)

< \-^—\ (• .-*.~.dA - {OiogxDîll 4 logo;] i J J2110B*) t [logx]i J

. f - e - « . « x I i t . ^ ] d i _ N - » ° " ' ( [ i o g 8 ] r [ — 1 J J l [ logs]; J

• [ V " " * " ( 1 + i ) [ l o « " d i < a-0-1, when 3/ Ss 62(1<"")-0'1.

Lemma 2. Set «(a) = e2"' and

M+N M+N

»=M+1 n=M+l

o„ being any real numbers. We let 2 denote a sum over all primitive characters

modulo q. Then it follows *

I M + N 12

S - f r S l S °-*«<n) <(X2 + *tf)Z, (2) i<x <p(q) x, !»=«+! <

and M+N

Z - r r S l S «^«|s«(e + f)z. (3) Proof. Let P be any complex-valued function with continuous derivative and

period 1. We average the inequality


| *(-§)|< 1*001+J JJWIItfl 1

over the interval 7(o, g) of length — centred at —. The intervals J(a, g) with 1 <

a ^ q, (a, g) = 1, and g ^ Q do not overlap. Hence we have

2 2 k ( f ) |<2 2 M,( in«)i<fa + -M .i*"0oi<v}

< <?2 P I P(«) Ida + -i- (' I F'OO l«V • Jo 2 Jo

Now put F(a ) = {S(a)}2. Since [' \F(a)\da = Z and

- M ' I *"(/») | d j » - P | S ( a ) | | £'(«)!<*« 2 Jo Jo

< { (£ |S(«)|2d«)(j[ |S'(a)|2d»)}4 - Z*'(£ |S'(a)|2d«)*,

therefore we have

ZS|«(f)| ,-SSi|*(7)ll'(-^^)ll"

- s s i s«({-(-+[f])ii)r l<4<q

= 2 2 2 a»+«+mefif)r IV-[|]

<ZQ2 + Z*{ 2 ((2-»>»+«+[-]V}4<^2

(4)

For each primitive character X, we put r(Xj) = ^ X*(a)e (—). We have r(Xj)-

•XJO) = 2 * ? < » (—)• Since |r(%*)|2 = q, we obtain

/ 1 \ * I M + N 12 / -. \ ' M+N i - >

(-7-7)2 2 «.w <(~h-)2*k^) 2 «.w

No. 2 CHEN: PBOOF OF (1 , 2) 161

(« ,9 )=1

(2) now follows from (4). Let h be a positive integer, satisfying the inequality 2*D< g < 2*+1.D. We have

* / , \ / „ I M + N I 2 \

i = 0 X & Dl n=M+l V i / ' »=M+1 = 0

Thus the lemma is proved.

Lemma 3. Let Z denote a sum over all primitive characters, modulo q. Set f-q -

S = a + ttf, and suppose that a ^ —. Then it is inferred that

£ 2 * lJ*s. x,)|<«Q2|s|2OogQ)4. « < 0 X ,

JProo/. We have

L(s, z) - 2 - ^ = 2 - ^ + 2 — ^ 2*«

i^*l1(£«°B-<rM-&: f1 iW + 0 (\S\qi\0gq\

D'

Using <r Js — and Lemma 2, we obtain it

, ,[01511 v / M< \

2 2* l (s, z.)i4« 2 2* ( 2 ^r 1 + o-'isivoogay) « < 0 t . 1<0 x„ X l » = 1 n ' '

101*11' « • N

«isi2o2OogQ)4 + (g2 + o 2 i s i 2 ) -S i L^2«e2 l«i2( iogo)4 . This proves the lemma.

Lemma 4. Let m be an integer, while m =H= 1. For square-free odd fc the estimation

| 2*Z*(m) | < | ( m - l , * ) |

holds.


Proof. For the proof we remark that if k = p ^ • -p ;, p, < • • • < pi, <fy is a primitive root mod p,- and, further, if m = gf'(mod p,) , 0 ,- Pj — 2, j = 1, • • •, I the primitive characters belonging to

fi(, 'id

modulo k are given by Z*(m) = e v ' ' . Set ZQm, k) =

have 2 J *fc(m) We

p—2 v f

m1 z(m, *> - n zo- a) - n s • 1

This proves the lemma.

Let a; be an even integer, Xx be equal to 1, and Xd be equal to 0 for d > a;4 T .

Set 0« = - i - , /(*)-*(*) n ^ . -d xt—-g&-{ £ *gf} <p0<0 fit P — 1 /(<OffW <• V , i /(*;) J

X - r f f o r ! < d < x* * • If ^ be an odd integer, ^(.d) * 0, then

ct,*>=i we have

(*) v ^2(fc) = v v "2(fc) = v J i i v ^ >

Thus we have | Xd | < 1 for all positive integers d. Denote by a; an even integer satisfying log x > 10'. We use the following notations:

<?= n p. Kf<x*

Q- 2 1, Jf- 2 (-^-\( 2 A(») •»<,,<,*<*«£)* A ^ W c - p ^ H ^ ^ A n<7k

We have Q < + N , where 1 - e

a;

PlPl (*-p,pia,0)=l

No. 2 CHEN: PEOOF OF (1 , 2) 163

I t now follows from Lemma 1 that

M< 2 ^—\ 2 A ( » ) t U - + 0 , * )

< s f-Vi s A(«)#(-i-)( s **y

+ ° (^)=) - (,§, J^ w ^ - ° fc^) • (5) 4i0 4i'Q

where

*„« - 2 (^-\ 2 A(«)*(-^-)

+ 2 *_ii£L<» 2 ulî ^ O . ^ W C T O H 1 0 ^ ,

A(t»)

I W W wT i) J

, ) Ut,d2)

z ««(**,)• M — U r ^ ; • <6)

We denote by *(d) the number of prime factors of d. For a character Xd, let d* be its conductor and X*t the corresponding primitive character. We use the following notations:

i ¥ , = y y — W i _ y /_L_Wwyp(—V

282

164 SCIENTIA SINICA Vol. XVT

V

1 , 1 N P i P ^ I xM<P,<.^<P 2<(f :)

T

Lemma 5. Let x be an even integer. Then we obtain .£?<;———- +0 ( — ) . 1 - e \(logaO ,-0V

Proof. From (5) and (6) we have

j r < j f l + | j f , l + j f 4 + o ( ^ F ) , (7)

where

M 3 = 2 2 -T4^T S (-^-W^—)' w,.»)=i «„»)~i T ( d,d, \ i ^ l t i V I P J » /

V \ ( d „ d2)/> -»<'.<*'<'.<(7l) ° V % p / P1P2

(<ili2.flf2")>l

*4- 2 2 I-—^ w,.»)=i Wj.»)=M 2izi P I P J

Now the s u m itf3 wi l l be estimated. W e have

*»«*• 2 i 2 ^ ) « 2 ( — ) ( 2 T « « • .»<*,<.*<*<(-*-,)* ."<*.**<*<(-?;)* d%

° < f l P 2

Now the sum Mt wil l be estimated. Set d > 0 and /i(.d) ^v 0. The number of

solutions of the equation ' 2 = d, where d, and d2 are positive integers, is given (d1; d2)

by 3"U). Since | 1 , | < 1, we obtain

f> P.P2

283

CHEN: PBOOF OF (1 , 2)

Since — <>, Xd) = E (a,, Z*.) + Y! *?•<>> lof P , it follows that

rz.p"'—*J»(p) d*

M,*^M2 + M5, (9)

L L ^7_ p"—*J„(p)

where

M ' _ V V(d) L^„ ' ( } J « - V» A dog.)'-1; J<x

( y X?,(p)logp\ y XîthPi) du) I

\ f£ P- ~ *WPV AP,<£,<(,L)* (ft*)" log £ ; ' ' ( ^ i . « = r " ' P'P2

the congruence £?/ = 1 (modd*). By Lemma 4 we have

W e have *iW = V (X**(P> Y when Re « = 2. Let j / denote the solution of

P- - « . 0 0 £ A p- '

1 2 * ^.OOX^CPIP**)] = I S*z"*(p>p*^|

 1, ft(d*) ^? 0 and (d*, xp^p*) = 1. From (10) and Lemma 1 we have

i <p{a) I d*[d ., i 1=1 h A / i \ 1 *J* w u v r •

<p,p2,d)=l

log / W M r *«r*"* S*>l',|-JV *"<P.<»*<P.<(-i-)* P1P2 W.*)=l <p,p2,<l)=l Pl

" c i p j '

t, ^ v , , - . 1 , * P ~ « * - • - " - ^ ? ( t , ) ? ( y i<j<(ioB_|-)(ioSp) > \jog—/ v.*v<**_! PIPJ '

S x S S «* - p«p*. *>» «*f S (* i .*)-t

* ' , » .*<,,<,*<,,«£.)* J O » *• «2=o(modp) » * < - £ -

f " P I P 2 »1»»

| 2 * 2 !«*'-• en) ^ di(x-PiPlpb Ki<xi- '

Prom (7), (8), (9) and (11) we complete the proof of Lemma 5. Lemma 6. We have

284


Proof. Set

Then it follows

j t f 2 <

• <*>(—,*,Wp,p2)| < £ IpWIa*"

\<«,*- <pC0 x< ,*<^*<,.«iotvoB—; «•")=» W,.<>-1 " PlP>

\m,m, / IJ

We have

where r ( l ) = £ 1. Hence we have

M2 « (log x)" Max JVm, (12)

where

P.PJ l o g J L l * * >

Let 5 J denote a sum extended over all primes p„ p2) satisfying x« -'(log x)m < **"« < 2''(loga;)100, and let I 2 be equal to [ 7 1 o g g 1 . We L 30 log 2 J

have

1 = 0 * = 0

where

A'S-w- S U W l ^ ' l<i<(l»g»)"" d

u.*)=l ct^ l o gJLJ Vft I J 2** (*."•> \ l o g

P I P J '

No. 2 CHEN: PROOF OF (1 , 2) 167

U0PI3"

for J > 1. ** <*•»•> U g - 5 - 1 PlPa 7

P.P2

Set SCff, w, XJ) = Y, ^ W ^ W _ w h e r e J J « j . p o r Re « > 1, we have

S(H, », X,) « logx, £(co, X„) = J ] b& + 0 ( | ( a | d * l o g d ) .

For He 1, CK(n) ==.— 2 i"(<0, so that C«(«) = 0 for d[n, 1 < H, and C„(n) = 0 for n > #2 . Applying Schwarz's inequality we obtain

WOWfOI1 , 3 ' i , 2 ' + 1 H

C«(M)X,(TO 1 (14)

when H « a;. When Q « a;, from (3), (14), and ^ r2(«) «; x(log x)3 we have

and

2 , +1H , l + l „

V _ J _ V * i V Cud'^WI'^fo | 2'ff\ V ««))* &., GO % I JL ^ ' ^ ^ JL ~^~

< < C ( 2 - | + i ) a 0 g a ; ) ' S —^T 2 * 11 - £(« + w, Z /SfCff, a + iv, Xd) 1

2

K< y _J__ y * | y C H O O ^ O O I2.,. la + t v p y g o g x ^

< < ( | + i + j ^ M l ) a o g l ) , (15)

where « = H . We have logx

—* <nP+'w (16)


where ft = 1 and | jQn) | ^ tin). 2 logx

From (3) and (16) we have

2 -^-^[SCB^ + i v ^ ^

« (2'(log * ) » + ^ ) 2 - ^ ^ « 2'(log x)'« + g 2 , (17) V 2'(log x)100/ ^r[ n 2'(log a ) "

1 f L(B X ) when J > 1 and H « x. Using £'(ro, Xj) = v s '—^- d£, where r is a circle 2jr4 Jr ( f — w ) 2

with centre to and radius (logx)-1, we have \L'(w,Xd}\ « ( logx)M | i(?> X^)|d|.

By Holder's inequality we have \L\w, Xd)\* « (logx)5 1 |L(f, X,,) |41 d | |. It follows

from Lemma 3 that

2 (-TJx) 2 V t f + *», X,)|4 « 2'(logx)'»'(|/? + « | ) ' . 2<-'(Iog*)100«<<2'(log*)"» V ' °°<*<2' ( log *) 1 C

XdCp&i)

' ( P l P 2 ) " l 0 g -

| l -£(co,X,)S(2r ,u>,X,) | ,

P1P2

2 ' _ 1 ( l o j x ) l » 0 « ( < 2 ' ( l o i » ) l » » d

2' X / P I P I ) •

«•-> cPlPly log -P I P J

| i '(o,,^)S(ff,co,^)|.

When J > 1, from (18) we have

J° | a + t - 1 , | ( i + J i ± i ? i L y " , i " + i

\ ( logx)1 1 / + x*f" B(l,k,p + iv,m,H) d

It is obvious that 3 log J

3><J) < ; e log log J

(19)

(20)

No. 2 CHEN: PROOF OF U , 2)

when | fi(d~) \ =N= 0 and d is large. We now treat N%,k) with three cases.

» I . . I . , U Case 1. Set I > 1. Suppose that I and fe satisfy 2 V > x2 or x2 > 2kx30 >

2'(logx)100. We put

H = 2'(logx)M07,,I, where /, . , - el°«'°«{2'<1<"">,0°} .

Prom (15), (17), (18), (19) and (20) we have

JV«-*'«x(logx )<("[{ £ - k ^ S ' l S ^(PlP;) I'! J° S'-l<l.«.)«»W(lo,:.)<l» « X,, '«.»•) (p ip2)«+." l o g - £ _ ' J

M. *>=! P1P2

^.y—1.. .i«i . . . . / . . ...inn d tj ' I > J 2'-l<lotx)™<4<2'(lojx)™ d *d

•(TT^)+a;i(iog^n( 2 i ^ s * i s V ! + v ' J" l V-i(io. ,>"»<^<(io, x,™ d », I a. «>

( P , p 2 ) ^ ' " l 0 g — V->( l„ e «)«»«<l '< lo ,* )«» rf *< ' p , p 2 u . 1)=1

• ( s -J-ÎTk*./»+«-. ^lYM^) V-i( io«*)>°»«<i ' ( io»*) i» d " J 1 1 / M + J»/

«x(iogxyn(2'(iog^)™ + - ^ - N ) ( 2 JLygqogx)" J»i\ 2'(iogxrA2tj<n<2t+1jw

2A ff + * , + (i + ^ ' d o g x ) ^ i*(_*^_) + x i ( l o g x).r- i ( n i o g , ) -

2'(logx)'°° H1 1 ' I \1 + W Jo IV

+ gi^)J'-4V^*)"+ff0«*>,,}^+^*(Tf^) * 0^0= • (21)

Case 2. Suppose that I and fc satisfy

2*x^ < 2'(log x)'°° < 2a;*-'.

Putting ff = max(x2 , 22'-*x~3° (log x)400/,,*), we have

iv-'<<x(1ogx)8[;{(niogxr + 2 ^ ) ( ias 3)

. f2'(log x)'°° 1 + ( l + v2)22'(logx)2°°\(J 0 * / dv \ V IT 2'(logx)10° ff2 / '•*yJ Vl + „ " /

+ xi(log x)< ["{ 2 -liS^lL 2 * |S(ff, /? + iv, Z,)|>}*tfi.,)* 2<-1<loB*>1 0 0<.*<2 ;Cloir*)™ » Xj


* . / i .. . i nn f . .inn (L v . ' V-l(loK*)im<d<2'(loB*)100 ^

2'-l(Io^)™<i<2l(IoE„100 d X„ l \ * . «) ( p ^ P + ' - l O g - ^ - ^ 1 + ^ P.P2

« xi(log x)2° {2'(log a ) " + - _ S _ - l i ( I / ] r ) * ( 2 ' ( l o g *)•»)* I 2'(log x)mi

. ( 2 i ( ] o g , ) , » + 2 ' * J \»f- (1 + vQ* d „ + » , « _ £ ^ . ( 2 2 )

V 2'(logx)I0V Jo l + „< (logs)20 (log a;)20

Case 3. Set I = 0, and suppose that 0 <l k < I2. Let c denote a positive constant, and Xd denote a primitive character. It is well

known that

L(S, Xd~) * 0,

when Re S S= 1 . Hence we have

JV«. *> « y j y M U ( d ) l v * | [ '"^ I 7 3 +"° v / 1

P1P2

M J . P / V (I02 x)1-1 / L w

«dogxr s (^_V""-^« *^. (23) , \»,iv (losr z)2° 1 1 , V.2V (log z)21

° < p , < * ° < ? 2 < (

From (12), (13), (21), (22) and (23) we complete the proof of Lemma 6. Lemma 7. Let 1 be a large even integer. Then we have

Mi < {(8 + 24,)sC«}{ v 1 log x I I , f-1

1

* < , , « * < * < ( ^ ) i ^ 1 ( « * P1P2

.here C, = ]7 E j = 1 I I f1 ~ T " ^ ) • f > > 2

PJ-OO/. Set S = 5 ] '"25fc? • Then it follows

(fc,*)=i

^ S / 1 , 1 / ( /(w)

Hence, if (TO, X) = 1, we have

No. 2 CHEN: PBOOF OP (1 , 2) 171

<*<(** *)* d<(x2 ) 2 1<*<(*"S )V<*

W ^ , i / ( r ) .171', Vd7 Sf(« l<r>a.x* )»

= ft(»0 m )

(r.'i)=l'

1 Since —- = gid^g^d]) V ) /(«!), therefore we have [ d,d2

x ^ \ ( d „ d 2)

2 E ,*'& x - S E i*,A (4)ff(4) E / «

E /co( S i^w)V--5- (24)

f*.**)"!1 tfd"(d,x]=i

Set 7*(a0 = V - ^ T • T h e n w e ° b t a i n

K » < » <p(»)

kg«<si< E n ( E ^ ) = E ^n( i - i )" ' = 00= E S ^ = E ^ E ^T<S^-^(«)

J I * t<^<* ») <it <P(«)

_ kVk(x)

Hence F f c(x) S* <p( f c ) l o g* h o l d s g 0 0 d. Set 0 ( 1 ) = 1 and 0 ( g ) = J J (p - 2 ) , where /c pi?

g > 2. Then we have

„ = y^ g'(fe) TT (-, . 1 \ = y e W y 1 ^ . . i *(*) M ^ v - v î.,1 «P(*) ^ 0(9)

J_<

= y ,»2(g) y ft2(r) > y ,»2(g) J«p(gg) lo,.s~4 2 l i_, i 0(3) <p(g) f i 4 *(r) " % t 0 ( g ) <p(g) l « * ° « J

?<C . . .

- (^ i ) aog x- ) n ( i+^) + 0 ( i )=^4^ + 0 ( 1 ) - ( 2 5 )

\ x I /-/, \ p(p - 2) / C* When a; is a large integer, from (24) and (25) we have

J f , < ( 8 + 24*)C,aogs)- S E (-Âoi—). (26)

*B<Pi<*3<Pz<(^7-)i "*P,P, ^ ° S p , p 2

X I ^PiP:)!


Lemma 7 is true from (26) and Lemma 1.

Lemma 8. Let x be a large even integer. Then Q <, —, —* holds. (logs)2

Proof. When a; is a large even integer, from Lemmas 5, 6 and 7, we have

^ ( 8 ( 1 4 - 5 e y c j { 2 ^ 1 (27)

We have

S — ^ - < a + o S

< (1 + 2e) ('* _**- ( ( P * * - < (1 + 2 e ) ( ^ p % ^ _ , 3 Slogs J , ( l o g 0 ( l o g iL ) k « J t /Ki-«-/»)kgx

J l a J | \1 — a A / J \—a—p J l a ( l — a)

i i ' + ' , / \ * * \ i • + 1 i 2 ~ 3 c t

J l a ( l - a) i ^ J1+1 o(l - a) ?r^ Ji+i a ( l - a)

«s{„(,,-ii)}L|=f-14 9 » 9 _1_ _»j 30 ~ 30J,

= 0 v ^ J-U' -* I JL__j *_ _1_ ,, J + 1 j 1 10 30 10 30 '

l ^ i + i + j , (• - (0.4 + 0 i - to) d a < S | ( 1 , _ 0 j ) + i ± i j {log 27^1

ttJl+Ji (1.6-0.O«(l-a) ^ i [ 16-iJl^3 + i

_ l o g 2 6 ^ i } _ 3 ^ ^ & ^ { 1 < e ( L 6 _ 0 , ) + l ± i l 4 -*- i J .^Jfo+i (1.6 - 0.0 (1 - « ) 1=0 l 1 6 - i J

. {log 108 + 2 3 t - ^ _ 3 £ ( ^ L _ Y l o g * L z ± ) I 78 + 2 3 i - i 2 J | ^ \ 1 . 6 - 0 . i A 2 6 - i /

< (0.47 + 0.25X0.32542) + (0.40547 + 0.33334)(0.26236) + (0.33647

+ 0.42858)(0.22315) + (0.26236 + 0.53847)(0.19671) + (0.18232

+ 0.66667)(0.17799) + (0.09531 + 0.81819)(0.16431)

I 015115 of0-03774 | 0-03922 | 0.04082 ,0.04256 , 0.04445 _3(M3J V lj 1.5 1.4 1.3 1.2

+ °-04652 + 0.04879 < 0.234303 + 0.193837 + 0.17073 + 0.15754 + 0.151115 1.1 /

+ 0.1501 + 0.15415 — 3(0.023587 + 0.026146 + 0.029157 + 0.032738

+ 0.037041 + 0.04229 + 0.04879) < 1.21178 — 0.71924 = 0.49254. (28)

No. 2 CHEN: PROOF OF (1, 2) 173

Prom (27) and (28) we complete the proof of Lemma 8. Let x be a large even integer and P x(x, XE) be the number of primes p satisfying

the following conditions:

p < x, p ?£ x (mod p,) ( 1 < i < | j),

where 3 = p t < p2 < • • • < Pj ^ X" is the set of odd primes not exceeding x&. For a prime p' , let Px(x, p', £») be the number of primes p satisfying the following conditions:

p ^ x, p = x(mod p ' ) , p ^ x (mod p,) , (1 2.6408x0, (log x)2

Proof. Putting r ( p ) = , K = x and z = xu in (2.11) of [11] , we see p — 1

that (A t ) and (A2) of [11] are satisfied. Therefore, applying (2.11) of [11] , we obtain

_ 1

<P(*) A* 1 _ J_ lo<r x" 1<>g X

P

¥>G0 M P(P - 2) M V (P - 1)V log xh I W x J

i M l + O p - ) } , (29) x i \log x/ >

20 e" log x I Vlog

where r denotes Euler's constant. Set

F( t t ) = — , / ( M ) = 0, 0 < « < 2 ,

(uF(u)Y = / ( « - 1), ( « / ( « ) ) ' = P ( « — 1), M S* 2. (30)

When 2 < u «S 3, from (30) we have

2e ' wF(w) = 2 F ( 2 ) , P ( « ) =

We have

« / ( « ) = [" P (* — 1 ) * = 2e r log (u — 1), when 2 < u < 4;

K „ ) = 2 « ' 1 o g ( « - l ) w h e n 2 < M ^ 4

M F ( M ) = 2er + ["f(t - V)dt = 2e' (l + ("""' I o g ^ ~ 1 ) d A , when 3 < « < 4;


5/(5) - 2e'log3 + ( ' F ( « - l)du = 2e' flog 4 + f — (" ' ^SL ^dt).

We put | 2 = rei-, « = 1 and s = ^ in Theorem A of [11]. Prom (2.19), (4.18) and (3.24) of [11], and (29) of the present paper, we have

p (a. J i ) > 2(1 - y~Qe- ra:C,/(5) -^ J8(l - V T ) xCx }

(logsXlogz") "X ^ x ) 2 '

. { l o g 4 + j ^ j p hga-D^}, (31)

where a; is a large integer. i-« i

Again we put | 2 = , a = p and « = a;" in Theorem 4 of [11]. Prom (2.18), V

(3.24), and (4.18) of [11] and (29) of the present paper, we have

S p^,P,xro)<fo(i+/o;-û s m i i *• ( l o g s ) 2 H , ^ - 1 , \ P /

^ 10 log P fa i

^ f(4 + 5VT)a;C ,JUfJ ' dS f4 - '^ log (t - 1) df

~~~\ logs JlJ,ft • / xi\ h t S(logS)(tog|-)

r * dg ] _ f(4 + 5VT)«C,1f f* da p - 0 " l o g ^ - l ) ,

' • ^ ^ • J ) ' " 1 «<*«>' "'Ml-.)1' • f* da I, = f(8 + loVT^xC, ) J l (1 V I (logs)2 J

V2 /

I J * 2 a ( i - - « ) j 2 * ' (32)

• ( * - )

By putting 4 — 10a = tt — 1, it is found a — and — = 10 Jl__„\ u(5-u)

We have

u = 4 when a = —; 10'

M = 3 when a = —.

293

No. 2 CHEN: PKOOF OF (1 . 2) 175

Hence it follows

f* da f4-10- log (t - 1) dt = f4 lOdu f""1

JI (1 \)i t J 3 M ( 5 — u ) ) i

H(t-Vdt t

We have

- l o g q - i ) ^ (4 ** r me*-!)* _ m f _d« r J3 u JJ # \4/Jfe 0 / l _ _ \ J J

_ (VJ, **—) du \ - ' i * ( « - ! ) * > (4 f M ^ J L I dtt

JaVtt 1i(5 — tt)/ Ji * J3 l«(5 — M)J

J2 V 2 t AtJ hl2u(5 — u)i\ w —1 /

J s \ 2 « 4(5 — M) U — 1/ 2 3 4 + 0.75 log •£- = J - + 0.75 log -^- - 1.5 log -i- - ^

2 2 8 3 4 > 0.588335 — 0.6048075 = - 0.0164725, (33)

be using

log x < ^ - ^ i + — — for 1 < a; < 2. 2 1 + s

From (31), (32) and (33), we have

r.(,.->-(|) Sff.C»»>»((-£ff-0 * » px(*,*h -(-i-) E p*(*> R **) - f -*"'

By (34) and Lemmas 8 and 9, we obtain

(logs)2

(34)


and Theorem 1 follows. By the same method as used in estimating Px(.l, 2) , we can easily find that

( 1 2 ) > 0 . 6 7 x C , Oogxy

and Theorem I I follows.

REFERENCES

[ 1 ] Eenyi, A. 1948 On the representation of an even number as the sum of a prime and of an almost prime, #36. AH CCCP, cep. MaT. 12, 57—78.

[ 2 ] Pan, Cheng-tung 1962 On the representation of an even number as the sum of a prime and of an almost prime, Acta Mathematica Sinica, 12, 95—106.

[ 3 ] 5ap6aH M. B. 1961 ApiMpMenmeCKHe OyroawH Ha peflKHX MHOJKecrBax, P,OK. AH I/3CCP, 8, 9—11. [ 4 ] Wang, Yuan 1962 On the representation of large integer as a sum of a prime and an almost prime,

Sci. Sin., H , 1033—1054. [ 5 ] IlaH HsH-ayH 1963 0 npcHcraBJieHHH qerHUX qnceji B BHfle cyMMbi npocroro a HenpeBOcxoflHiuero 4

npocrux npoH3BeaeHHH, Sot. Sin., 12, 455—474. [ 6 ] Bap6aH M. B. 1963 lljioTHOCTb HyjieB L-pswoB flnpaxjie H aaaaia o cnojKeHHH npocrux H noTra

npocTbix iHceji, Mm. c6., 61, 419—425. [ 7 ] Byxurra6 A. A. 1965 HoBUe pe3ynbTaTbi B HccneflOBaHHH npoOJieMM rojityi6axa-3ft.nepa H npoftneMU

npocrux qHcan 6jiH3HeuoB, ROK. AH CCCP, 162, 739—742. [ 8 ] BHHorpaflOB A. H. 1965 O nnoTHOCTHOft rHnoTeae fljis L-paflOB Anpnxjie, Haa. AH CCCP, cep. MaT.,

29, 903—934. [ 9 ] Bombieri, E. 1965 On the large sieve, Mathematika, 12, 201—225. [10] Chen, Jing-run 1966 On the representation of a large even integer as the sum of a prime and the pro

duct of at most 2 primes, Kexue Tongbao, 17, 385—386. [11] Eichert, H. E. 1969 Selberg's sieve with weights, MathematOca, 16, 1—22.

A New Mean Value Theorem and its Applications

PAN CHENG-DONG

Mathematics Department, Shandong University, People's Republic of China

1. INTRODUCTION

Let MX ;d, I) = I 1.

pml(mo<id\

In 1948, A. Renyi [16] proved the following Theorem:

THEOREM 1. For any given positive A > 0, there exists a positive r)>\ such that

i r ( y ; l , l ) R(Xr';X) = I max max

d « X " y*X ( l ,d)- l i r (y ; < / , / ) -

•AW) « X log~AX

where 4>{d) is Euler's function.

Precisely speaking, the result of A. Renyi was proved in a weighted form, but the elimination weights did not present any basic problems.

By this, he proved the following proposition:

Every large even integer is the sum of a prime number and an almost prime number with the number of prime factors not exceeding C.

For brevity, we denote the above proposition by (1, C). Renyi did not give the quantitative estimate of 17 and C. By his method, we can say only that 77 is very small and C is very large.

276 PAN CHENG-DONG

M. B. Barban in 1961 [12] and I in 1962 [5] proved independently that Theorem 1 holds for TJ < g and TJ < 5 respectively. With TJ < 5 I first proved the quantitative result—(1, 5). In 1962 Wang Yuan proved (1,4) using only TJ < i I in 1962 [14] and M. B. Barban in 1963 [13] proved independently that Theorem 1 holds for 17 <§ , and we obtained (1, 4) without much numerical calculation. In 1965 A. A. Buchstab proved (1, 3) by use of TJ <§.

In 1965 A. I. Vinogradov and E. Bombieri [1] proved independently that Theorem 1 holds for TJ < 5.

More precisely, E. Bombieri proved the following important theorem:

THEOREM 2 (Bombieri). For any given positive A > 0, we have

7 r ( y ; l , l ) £ max max rr(y,d,l)-

<t>id) « X log"* X

where Bi = 3A+23.

From this, (1, 3) can be derived without much numerical calculation. In 1975 Ding and I proved the following new mean value theorem:

THEOREM 3. Let

7r(X;a,d,l)= I 1 ap«X

ap^l(modd)

and let f(a) be a real function, f(a)« 1; then, for any given A > 0 , we have

ir(y,a, 1, 1) V max max £ f(a)[ir(y; a, d, I)

«X1 /riog"B2x y * x (W)-l a<X1"' V 4>{d) (a.d)=l

« X log""4 X

where B2 = lA + \l and 0 < e < l .

Putting

/(fl) = l0) a>l we have

v ti \( 1 J ;\ ^ ( y ; « . 1. 1)\ , . , , 7 r ( y ; l , l ) alx_na)[rr{y;a,d,l) ^ _ ) = . ( v ; rf,/)--___,

(a.d) = l

so that Theorem 3 is a generalization of Theorem 2. However, its interest is less a matter of generalization than of important applications. We give some examples in Section 3.

18. A N E W M E A N V A L U E T H E O R E M

2. THE PROOF OF THEOREM 3

In order to prove Theorem 3 we require some well known lemmas.

LEMMA 1. For any complex numbers a„, we have

277

and

y " y*

1 „ .

M+N I M + N I anX(n) \«(Q2 + N) I \an\

n - M + l I n = M + l

M+N I / \T\ M + N

I anX(n)U[Q + -) I \an\\ n - M + l I V t i l n=M+l

H<q*iQ<f>(q) * ,

where the asterisk indicates that the sum is taken over all the primitive characters mod q.

LEMMA 2. IfT^l and \cr-\\*k 1/(200logqT), we have

and

Let

and let

I * f \L(o- + it, X)\idt«<t,(q)T log4 qT X, J-T

I * f | L V + it, x)\4dt«<j>(q)T log8 qT. x„ J-T X« J~T

V(X;a,d,l)= I \(n)

an •/(mod d)

R(D;X,f) = £ max max iHy,a,d,l)-< A ( y ; q , l , l )

(a,d)- l where

D = A7 log - " 2 *, B 2 = \A + 17.

For (a, d) = {l,d) = \, we have

<t>\d)an<.y <P(d)x4*X% anty

log d log y\

I., *™+°(rH$i) (n.d)=\

278 P A N C H E N G - D O N G

From this, we have

R(D\X,f)* I — r I m a x l * I f(a)x(a) I A(n)*(n) d*D<t>{d) l « j | d y « X x, l a * X '

(a.<i)=l <n.d)=l

+ o f - ^ — ) = £ log AT max I - f r l *

x I / (a)*(a) I A(n)*(«) l a s X 1 —

(a.m)=-l an<y

(n,m)=l

Let h be any fixed positive number and £>i = log X. From (1) and the Siegel-Walfisz theorem, we get

R (D, X, f) *£ log X max I - ^ max I * i D D , < « « D ^ ( ( f ) y « X *,

I f(a)x(a) I A(n)*(n) a « X ' " anly

(n.m)=l ° ( i o ^ ) - (2)

Let £>i« Oi ss D, Q < Q' *£ 2 0 and let (q) denote the interval Q < </ = <?'.

Let j « E < A"' ', E < E' *£ IE and let (a) denote the interval E < a =s E' . Let

I m ( 0 , E ) = I - f - m a x I * ( < l ) 0 W ) y * X x. (a) an<y

(a.m)»l (n,m)"l

It is evident that Theorem 3 follows at once provided

I m ( Q , £ ) « A+3

log X (3)

For convenience, let

/"o-l*; / ( a ) , (m, a) = l ,

(m, a ) > l , and

and let

dt\n) = \{n), E^D\,

\\(n), ( « , m ) = l ,

(«, w ) > l ; dT(n) \ 0, E>Di

1 Im' (0 ,E) = I — - m a x l * I /< m , (ak( f l ) I <4m> («)*(«)

W0\R) y « X « , Ma) l n < y

18. A NEW MEAN VALUE THEOREM 279

Then we always have

Im' (Q, E) = Im' (Q, E) + o ( l p * ^ • (4)

By Perron's formula we get

I m ' ( 0 , E ) « I - f - m a x I *

x l C / S ° h *>*,(**> 'ids\+°(i=^f) (5)

where

5 = o- + i7, 6 = 1+—^—, r = X 1 0 , l o g *

d£\s,x)= I ^ " ' ( B ^ I I ) " " , c r > l ,

f<EHs,x)=I.f<m)(a)x(a)a-\ (a)

LEMMA 3. If E ^ D] we have

Im'(Q,E)«XDr \ogl3X + XiDDi\ogbX. (6)

Proo/. Let Mi = QD i and

and, for brevity, let G, F and H denote d ^ U * ) . / E " ' ^ , x) and H(s, * ) . Then

FG=FG(\-LH)+FGLH = FG{1-LH)-FL'H. (7)

We have

F G = I a(n)x(n)n-'=F1+F2, (8)

with

^ i = I a(n)x(n)n-\ F2= I a(n)x(n)n-s. (9)

where

/In \ / /

2 8 0 PAN CHENG-DONG

From (7), (8), (9) we have . t + iT

f FGy-ds=\ FGy-ds=[ F2(\-LH)y-ds

+ f (Ft-F^LH-FL'm^ds + OiX-1). J(i.D S

From this, by Schwarz's inequality, we get

Im' (Q, E)« X log AT max ( I - ) - 1 * |F2|2) Res-i \(«)<PW) x„ I

l

x m a x f l - i - r l l - L / f f V Rcs-b Vn)<PW) x„ I

+ xhogXQ* max ( l - ) - £ * |F,|2Y

l

+xhog^Xmzx(l-^-l*\F1\2)i

Res-* V(<j)<P(<7) *, /

Res=i V<7> <M<?) *, / \<7><M<?) *. Jtf.T> |5| 7

+ XMog^X max ( l - J - Z * |F|2) max. ( l - £ - 1 * | / / | 4Y

xflTT-rrT Trl*l)*. (10)

By using Lemmas 1 and 2 to estimate every term of (10), we can get (6) at once.

LEMMA 4. / / E > D\ we have

Im'(0,H)«AX>r1 \otfX+X*D\o£x. (11)

Proof. Taking M2 = Q2, when Res = b = 1 + l/(log A"), we have

G = d£)(s,x) = Gl + G2,

and

f F G ^ d 5 = f FG2^<fr+f FG.^r f i + O ^ - 1 ) . J(6.T) 5 J(A,r) 5 J (J,r) S


From this, by Schwarz's inequality we get

Im' (O, E)« X log X max ( I —— £* |G2 |2) Res = <> \ ( , ) 0 W ) *„ /

a x ( Z - f - I * | G | 2 ) 2 + xhogAT

x f X - ^ - r i G . l ^ m a x f l - J - r l F l 2 ) 2 . (12)

x max Res =

x max Res*

Similarly, by using Lemmas 1 and 2 to estimate every term in (12), we get (11) at once.

Choosing ft = , 4 + 16 from (6), (11) and (4) we get (3) at once, and Theorem 3 is proved.

Remark. If /(a) satisfies conditions

I \f(n)\« X logA' X, I I \f(d)\« X logx*X, (A)

where Ai, A2 are positive constants, then Theorem 3 is still true (B2^ g(A,A,,A2)).

3. APPLICATIONS

A. To the result (1, 2). In 1966 and 1973 Chen devised a new weighted sieve method and proved

(1, 2). Chen's principal contribution is that he pointed out that the key to proving the Proposition (1, 2) is to estimate the sum ft

ft= I 1, (Pl.2>

P3*WP1P2 N-p = PlP2P3

where N is a large even integer, and (pi,2) denotes the condition Nw<pi < N*^p2^(N/p1)

i; and he was the first to propose a method to estimate the sum successfully. In 1975 we pointed out that the key to realizing Chen's weighted sieve method was precisely Theorem 3.

Let P = ri2<p*Ni-/2.p|jv; t hen we have

(Pl.j) P*WP1P2 (/V-p,p2p3 .P)=l

I A„] HN-PIP2P3.P) >

+ 0(Ni)

+ (<*.N)-1 (p\Pi.d) =

)|)+0(AP),

282 PAN CHENG-DONG

where kd are the Selberg functions (Aj = 0, d>N*~'n). Hence we have

[l^l I \dl\d2 I ir(AT;p1p2,[</1>rf2],A0 + O(Arl) di\Pdj.\P ( p , . 2 )

<- V V V i i ?r(W;PiP2. 1. 1) 5 5 L 2- 2, ^di^dj iirj J 1\ (P1.2) rfii/*rf2l/* 0 l L « i . " 2 j ;

o( I U(rf)|3"W) I U(N;Plp2.d,N) \<l«JV"/2)— (P1.2) \

) | ) +0(N J )

<- V V V 1 i 7r(^;PlP2, 1, 1) « Z, Z. 2- AÂ^ ——————

(PI. 2) d^p d2\p <PK\.au"2l)

+ o( I |MW)|3"W ) I f(a)(n(N;a,d,N)

("'N>"1 ir(AT;a,l,l)

where . JL i / / V V 1, fora=pip2 , and JV1 0<p,sJV5«p2^ — ,

[0, otherwise.

Therefore it follows from Theorem 3 that

n =£ principal term + O (TV/log3 N).

B. The upper bound of D(N)

Let D(N)= I 1.

In 1949, A. Selberg proved

D(A0«16(l + o(l)G(A0—¥rr. log W

where

3 ( N ) = n ^ n (i-r-1-^)-p l ^ p - 2 p > 2 \ ( p - 1 ) /

In 1964, using Theorem 1 with TJ < j , I improved the coefficient 16 to 12 [15]. Until 1978, the best result was due to E. Bombieri and H. Davenport [2] who improved the coefficient 12 to 8 as early as 1966.


It is very difficult to improve the coefficient 8. In 1978 Chen [4] improved the coefficient 8 to 7-8342, but his proof is very very complicated. Recently, Pan Cheng Biao gave a simple proof of Chen's result. He proved the following:

Z>(JV)«7-928©(N)—^-. log N

I am going to sketch his proof. Let S8 = {b = N - p, p < N}. It is easy to see that

D(A0«S(flB,P,Ar4) + O(JVi), (13) where

S{98,P,z)= I 1 be3t

(6,pU))=l

and

9 = {p:pXN), P(z)=Y\p. pe.9 p<z

By the Buchstab identity

S(» ;P ,2 ) = S ( » ; P , w ) - I S($p,0>,p) (14) pe9

where :5>w^2 and $}d = {be 38, d\b}. It is easy to prove that

S(»;P,Af i )*S(»;P,Ar*)- ini + £n2 + 0(N') (15) where

H,= I 5 (» P l ;P ,An , (16) N , / 7 « P l < N 1 / 5

and " 2 = ,„ £ , S(S3P1P2P3;<?,P3). (17)

By the Jurkat-Richert theorem [11] and Bombieri's theorem we can get

S(fl8;0\AP)-2fti .2-5 ,

; 8 ( 1 + 0 ( 1 ) ) S ( M _ - _ [ 1 + | ; lafizlS^

. , f " l o g ( 2 . 5 - 3 . 5 / ( , + l»J ( 1 8 )

Jl-5 ' J

However, we cannot use the same way to estimate the upper bound of Q2 because in this case, max piPiPi^N*.

284 PAN CHENG-DONG

For estimating Cl2 we have to consider the set

<e = \l = N-(np2p3)p1;N°i^p2<p3<N*,\^n^—3, I P2P4

(n)î)) = 1;,3<Pl<rnin(M^L)). V p2 I \ npoPsJ >

It is clear that n2« 1 1

so we can get

n 2 « S ( i ? ; 0 \ A ^ " E ) + O(A"). (19)

When we use the simplest Selberg upper bound sieve method to estimate S(i?; 2P, N3~e) the error term can just be estimated by using Theorem 3 but not Theorem 2; and then we get

5 ( i ? ; ^ , ^ 1 - ' ) « 8 ( l + o ( l ) ) S ( N ) r ^ - (20) log AT

where X= I I I . (21)

N 1 / 7 s p 2 < p 3 < p l < A / 1 / 5 l «n*N/p 1 p2P 3

By the Buchstab asymptotic formula

I 1= :—£T77<U(U) + O ( - U , (22) i<n*y logy $logy ') /

( n . P ( y , / u ) ) = l

w(«) = - , 1 * £ H < 2 u

,(WW(«)) = W ( M - 1 ) , U > 2 ,

we can get

1 w ( u ) < T ^ 6 l ' M > 2 -

From this and (22) we have

X < l 4 ( 3 1 0 ^ - 1 ) ( 1 + o ( 1 ) ) l o ^ - (23)

From (23), (20), (19), (18) and (15), we have

S ( 3 9 ; ^ , A r * ) < 7 - 9 2 8 W ) — £ 7 7 ; (24) log N


and from this and (13), we obtain

P ( A T ) < 7 - 9 2 8 S ( N ) r - ¥ -log N

C. A generalization of the Titchmarsh divisor problem. It is well known that, by use of Theorem 2, we can get the asymptotic

formula

I d(p-l)~dX

where d(n) denotes the divisor function, and C\ is a positive constant. Using the new mean value theorem, we can get even the following result:

Let ls£y *£X1~e{o<e <1) , and let f(a) be a real function satisfying the condition (A); then we have

I f(a)d(ap-$~2X I - T J ^ I nJ^l/ny ap*x d*xU2{d)a*;ya\og(X/a)

Putting •1, fl = l ,

we obtain

/ (a) = to, fl>i.

I d{p-\)~CxX. p«X

D. The largest prime factor ofp + a. Let Px denote the largest prime factor of

n (p + a) oXe when 0 < §. The key of his proof is

the estimation of the sum

V(y)= I logq (25) p+a=fcq

y<q«ry

where q denotes primes, and X s < y < A"*, 1 < r < 2. Using the Selberg sieve method, we can turn the estimation of (25) into

estimating the following sum:

X I I log?. d*xul\o&-Bx k*x'y kisX

kq =a(modd)

It is clear that our theorem can be used here, too.

286 PAN CHENG-DONG

Now I am going to give a brief explanation of the relation between the sieve method and the new mean value theorem.

Let N be a large integer, ^ a set of positive integer satisfying the conditions

{e,N) = l, o<e<x1~v>, o < r , , < l , e e ? , and let

<e = {l = N-ep, ee%, ep^N}

9> = {p:p*N}.

Evidently, when we estimate the sifting function

5(i?;SB ,2)= I 1, z^N^''2, o<e<\. (26)

</.PU»=l

By making use of Selberg's sieve method, the error term can be just estimated by the new value theorem provided

/ ( « ) = I 1, e = a

satisfies the condition (A). It is well known, that before Chen's work, we could not estimate the

following sum of sifting functions,

I S(SB,;0»„z,), (27)

1

when m&xq^N1, where 3. is a set of different positive integers, 39 = {b = N-p, p<N}, SBq = {be3S, q\b}, §>q is a subset of S9 depending on q, and zq is a positive integer depending on q. Because when we used the Jurkat-Richert theorem to estimate every sifting function S(88q; (Pq, zq) the total error term caused by every 5(S9„; 9V zq) could not be estimated by Bombieri's theorem; of course, we can estimate the sum (27) under Hal-berstam's hypothesis.

Chen was the first to devise a method to estimate some kinds of sums (27), when iV^raaxâ i j ^ JV 1 "" 2 , 0 < T J 2 < 1. Briefly speaking, the idea of his method is to turn the estimating of the sum (27) into estimating (26); and we pointed out that the key to realizing Chen's method is just the new mean value theorem.

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