View
1
Download
0
Category
Preview:
Citation preview
MATHEMATIKAA JOURNAL OF PURE AND A P P L I E D M A T H E M A T I C S
VOL. 27 PART 2. December 1980 No. 54.
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE
E. FOUVRY AND H. IWANIEC
§1. Introduction. The celebrated theorem of Bombieri and A. I. Vinogradovstates that
, maxd/2)-e ( a , a ) = 1
n(x;q,a)-n(x)
4x(logxyA, (1)
for any e > 0 and A > 0, the implied constant in the symbol <g depending at moston E and A (see [1] and [14]). The original proofs of Bombieri and Vinogradovwere greatly simplified by P. X. Gallagher [4]. An elegant proof has been givenrecently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10]and H. -E. Richert [12]. Estimates of type (1) are required in various applications ofsieve methods. Having this in mind distinct generalizations have been investigated(see for example [15] and [2]). Y. Motohashi established a general theorem which,roughly speaking, says that if (1) holds for two arithmetic functions then it also holdsfor their Dirichlet convolution; for precise assumptions and statement see [11]. Sofar, all methods depend on the large sieve inequality (see [10])
I I*q i Q *(mod q)
(2)
which sets the limit x1/2 for the modulus q in (1) and in its generalizations.It is the aim of this paper to present arguments which yield theorems of
Bombieri-Vinogradov type with an extended range for q. We shall treat carefully
n{x,z;q,a)= £ fz(n),n ^ x
n = a(modq)
where fz is the characteristic function of the set of integers n having no prime factorless than z. Let us introduce also
n{x ;z,q)= £ fz(n).
[MATHEMATIKA, 27 (1980), 135-152]
136 E. FOUVRY AND H. IWANIEC
We have proved the following
THEOREM. Let z < x1 / 8 8 3 and 1 < \a\ < x. Then, for any A > 0,
I n(x, z ; q, a)-—--- n(x, z ; q) (3)
the implied constant depending only on A.
Our method applies to a wide class of arithmetic functions f(n), for which the
sum
nf(x;q,a)= /(")n < x
n s u(mod q)
can be rearranged as a sum
lm = a (mod <y)
(4)
of bilinear forms, with the variables of summation / and m in appropriate intervals.Such a representation for fz(n) is obtained through a combinatorial sieve identity(see Lemma 1). We failed to obtain (3) in the most interesting case z = x1/2, in otherwords for f(n) = A(n). In the latter case, Vaughan's identity (see [13]) would serveas a bilinear form (4), but unfortunately with L and M not well enough controlled forour method to apply.
Acknowledgement. The authors express their gratitude to Professor H.Halberstam for pointing out some errors and for some helpful remarks about thefirst version of this paper. Our work was done when the second author enjoyed a oneyear visit to the University of Bordeaux I. It is a great pleasure for him to take thisopportunity to speak of the pleasing scientific atmosphere in which collaborationwas so fruitful.
§2. Sketch of the main ideas. The bilinear form (4) is approximated by
(lm,«) = 1
with the total error less than
R(M,L;Q)=Q<q<i2QM<m<i2M L < I sS 2L
(q,a) — 1 ("i,g) = 1 lm = a(modq)
a, -L < I si 2L
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE 137
The problem of bounding R(M, L ; Q) is reduced, by the Cauchy-Schwarz inequality
R(M, L;Q)^ {QM)ll2D1!2(M, L;Q),
to that of bounding the dispersion
( I I Y(q m \ L < I ^ 2L /
Im = a(modq) (/, q) = 1
= W(M,L;Q)-2V(M,L;Q)+U(M,L;Q),
say. Each of the terms U, V and W is evaluated separately, the most difficult beingW. By definition
W(M,L;Q) = E Z Z «/,<VQ < ^ 2 Q M < m ^ 2 M L < / i , / 2 ^ 2L
(y, a) = ! (m, q) = 1 l\m = /2m = a(mod^)
With an admissible error we may replace W(M, L; Q) by W*{M, L;Q), whichstands for the same sum with the range of summation restricted to (lx,l2) = 1>lt = /2(m°d <?)• In particular the diagonal terms lt = l2 disappear.
When treating W*(M, L; Q) we carry out the summation over m first. We use Jx
to denote the reciprocal of /j modulo g, so that Ij^ = \ mod<j. Writing
M
M < m « 2Mm = al, (modq)
it is trivial that \r(q, afjl ^ 1, but this turns out to be not satisfactory. We obtain agreat cancellation of the errors r(q, a l j in the sums over lt, l2 and q. By expandingeach r(q, al^ into a Fourier series, a typical term to be considered is
Wh(M,L;Q)=< , s ; 2 e L < / i , l 2 « 2 L
IJ,<I) = 1 / i s /2(modij)
with li f 0. Since Q will be nearly as large as L and /j = /2(mod q) there is not muchroom for summation over lt and l2. For this reason we reinterpret the condition/, = /2(mod q) by writing
h-l2 = qr with 0 < |r| < L/Q, (r, l,l2) = 1 .
Here r is r a t h e r smal l , so t h e c o n d i t i o n lx = / 2 ( m o d r ) c o n s t r a i n s t h e va r i ab l e s llf l2
less t h a n d o e s lt = I2(modq). In a d d i t i o n ,
r U
138 E. FOUVRY AND H. IWANIEC
Therefore we arrive at sums of the type
ZZ0 < \r\ < L/Q
( l l , / 2 )
a,,a,2e( -ahr~),
with some Lt e (L, 2L], the factor
being removed by partial summation. A connection with the incompleteKloosterman sums is suggested. By the Cauchy-Schwarz inequality,
( l . / 2
Using Weil's estimate of XJ one just fails to get a non-trivial bound because themodulus /' /" is as large as the square of the length of the incomplete Kloostermansum Y,i- Hooley's conjecture R* (see [7]) would be helpful. In order to avoid anyunproved hypothesis, we appeal to a particular property of the coefficients a, torearrange the sum Xi,,/2
m t o a n° ther bilinear form with variables of summation of adifferent order of magnitude. Then, the above procedure yields incompleteKloosterman sums which are manageable by Weil's estimate. We doubt whether theelementary result of Kloosterman [9] is sufficient.
From the main terms in the dispersion D(M, L; Q) we get
M X <t>~2(i) ZQ < q ^ 2Q X (mod q)
101,-
L < I < 2L
We estimate this by applying the large sieve inequality and the Siegel-Walfisztheorem in a way familiar from the Barban and Davenport-Halberstam theorem.
§3. Lemmas. Let P{z) = H P < Z P for z $; 2. Let F(n) be an arithmetic functionvanishing for almost al t«. By the Buchstab identity
X fz(n)F{n) = X F(n) fP(n)F{n)) ,p<z \n = O(modp)
on applying the 'exclusion-inclusion principle' familiar from combinatorial sievetheory (see [5] and [8]) we obtain
LEMMA 1. Let D 7z z ^ 2. Then
E/,(n)F(«)=n 30(modd) X
d \ P(z)Z
nsO(modd)
ON A THEOREM OF BOMB1ERI-VINOGRADOV TYPE 139
where p(d) stands for the least prime factor of d, and, for a square-freed = pj ... pr > 1, px > ... > pr, we define
[ ( - I f , ifpl...Plpl<Dforalll^r,
{ 0, otherwise,and
( — l)r, if p^ ... p,p, < D for all I < r and px ... prpr ^ D ,
0, otherwise.
For d = 1 we define Aj = 1 and cr1 = 0.
Note that if ld j= 0 then d < D, and if ad + 0 thenobtain the
d < D. Hence we
COROLLARY. Lef D ^ z ^ 2. T/zew
Z /x(n)F(n + E Ep < z Dip1 Hd < D/p
pd | P(z)
X fp(l)F{dlp)
The following result is known in sieve theory as a 'fundamental lemma' (see [5]).
LEMMA 2. For R, z > 2 and (a, q) = 1 we /iat>e
E /.(«) = - n ( i - l N ) {n = a(mod(j) p|</
where s = logR/logz. T/ie implied constants are absolute.
LEMMA 3. If x(q) is the number of divisors of q,
nix 1(n,q) = 1
LEMMA 4. Let ip(^) = £, — [^]— ^ and A > 0. There are two functions A(£) andperiodic in £ (mod 1) such that
h + 0
B(h)e(hH) + A
withh ± 0
E. HJUVRY AND H. IWANIEC
and e(z) = e2nt2.
Proof. Take functions A(£) and B(£) of class C3 whose graphs are
n n+l*- A->
and whose derivatives satisfy Aip)(£), B(p)(£) <? A~" for p ^ 3, (compare with Lemma2 of [3]).
The next lemma is a consequence of Weil's estimate for Kloosterman's sums. Theproof is similar to Lemma 3 of Hooley [6].
LEMMA 5. Let 0 < A2-Ai ^ b. Then
A, <a a A2{a,be) = 1
a = A (mod A)
e(dr (b,d)ll2bll2T(bc)\og2b.
The implied constant is absolute. The notation a used when writing a/b or in acongruence (modi») means that ad s l(modfr).
LEMMA 6. For any pair a, b of non-zero coprime integers,
LEMMA 7. If x is a non-principal character modd, and d ^ ( l o g ^ , then
for any A > 0, the implied constant depending only on A.
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE
Proof By Buchstab's identity,
E X(n)fz(n) = E Z(«)- E X(P) E X(n)fp{n)P < z n <i i
where zx = min(z,obtain
141
I X(n)fp(n)
. Letting R = £1/2 in the 'fundamental lemma' we
X(n)fP(n)n <i UP
= II (modi) P I <
PI i J
Pi
logi?
where sp = logR/logp ^ logK/logZj ^ ^(log^)1/2. The second double sum is emptyif z < exp (yAog £); thus we assume that z > exp (>/log <) = zx, and we obtain
E X(n)fp(n)= E Z(«) Ei H/p n < {/zi zi < p =S rain (z,{/n,p(n))
« £ nn s£ £/zi '*
by the Siegel-Walfisz theorem. This completes the proof.
COROLLARY. Under the same assumptions,
E y(n) fJn) <
( n , e ) = 1
§4. Reduction of the problem. We split up the sum (3) into < (log x)2 sums oftype
E /Z(«)-TT3 E /,(«)S(y, Q)= EQ < < ? « ; _ .
(q,a) = 1 n = a(modq) (n,q) - 1
with 2y sg x and 2g ^ x11/21. It is sufficient to show that
S(y,Q) <x(\ogx)-A-2,
for z s£ x1/883 and 1 ^ \a\ ^ x. We have trivially that
(5)
E. FOUVRY AND H. IWANIEC
so that (5) is obvious for y < x(logx) ^"2. In what follows we assume that
x(logx)- < y ^ x. (6)
Now we want to rearrange S(y, Q) as a sum of bilinear forms. For this, apply Lemma1 twice to the characteristic function of the set of integers n e (y, 2y], n = a(modq)and to the characteristic function of the set of integers n e (y, 2y], {n, q) = 1. Thensubtract l/<t>{q) times the second inequality from the first, to obtain, as in thecorollary to Lemma 1,
Z L(n) -y <n ^ ly
n = a(mod</)
L(n)y <n ^ 2y(n,i}) = 1
I 1-Id < D y < n ^ 2yd\ PU) " = "(mod^)
(d,q) = 1 n = O(modrf)
y yn = O(modd)
+ z z fPin) ~p<z Dp~2 ^ d < Dp~^ y < n ^ 2y
pd \ P{z) n = a(modq)(pd,q) = 1 n = O(moddp)
y<n^2y "n = 0 (mod pd)
() i
p <zpit
say. Hence, in the above notation
Sp(q,D)< z g < <J s: 2(2
() 1
(7)
say. The sums Sp(y,D,Q) with p < min(z, exp(%/logx)) = z0, say, will beestimated easily by means of Lemmas 2 and 3 while those with larger p will betreated by a dispersion method.
§5. Estimate o/S^j;, D, g). For (d, q) = 1 we have
Z
and, by Lemma 3,
y < n <i lyn = a(modg)n = O(modii)
Zy<n(,ly
n = O(modii)
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE 143
Hence S^q,/)) -4 D and consequently
Sl{y,D,Q)<QD<xi~u, (8)
provided
QD^x 1 " 2 8 , (9)
which we henceforth assume.
§6. Estimate of^,p < z0Sp(y, ^> 2)- By Lemma 2, for each a with (a, q) = 1, wehave
V /•(«) = — ITpmn s a (mod^) pi | q
where i? is any number ^ p and sp = logi?/logp. Hence
Dp-i < m a DP-I \pqm j q p
and consequently
P P
For /? = xE/2 this bound yields
^ Sp(y, D,Q)<x exp ( - (log x ) 1 / 3 ) , (10)P < Z0
the implied constant depending on e only.
Now we proceed to estimate Sp(y, D, Q) with z0 ^ p < z.
§7. Rearrangement of Sp{y,D,Q). Let M take the values 2"'Dp~1 forA = 1, 2,... such that Dp~2 < M < Dp"1, so that there are at most 21ogp such M's.We split up Sp(y, D, Q) into < logp sums of the type
EJy, M,Q)= ^ ^ ^ ' - 1
thus obtaining
< pmn ^ 2y ^rXH/ y < pmn ^ 2j{q, ap) = 1 (m, q) = 1 pmn = tj (mod g) (n, pg) = 1
(P, n) = 1
, B, Q) X ^ P ^ ' M - 2)+ 0 (wr 2 log x). (11)M
144 E. FOUVRY AND H. IWANIEC
Here, the error term comes from the contribution of rc's divisible by p2. This error isadmissible because
X yp ~2 log x < y(log x) exp( - ,/log^x).zo sg p < z
§8. The dispersion. By the Cauchy-Schwarz inequality we obtain
E2p(y,M,Q)^MQDp(M,Q),
where Dp(M, Q), called the dispersion, stands for
DP(M,Q)= X I ( I / P W - ^ ^ /p(n)T(ij,ap) = 1 (m, q) = 1 pmn = a{modq) {n,pq) = 1
= Wp(M,Q)~2Vp(M,Q)+Up(M,Q),
say. Each term Up, Vp and Wp will be evaluated separately. By definition,
UP(M,Q)= £ X ( J _ S /p(n)YQ < i } 5 ; 2 g M < m « 2M vPVH) y < pmn 2y(g,aj>) = 1 (m, g) = 1 (n.P«) = 1
§9. Evaluation of Vp(M, Q). By definition,
^ /p(»l)/p(»2)g < q s: 2Q V l y J JV/2 < ni ,n2 < 2JV Mi < m < M2f(j,ap) = 1 {n\n2,pq) = 1 m apn\{modq)
1/2 < ni/«2 < 2
where for simplicity we have written JV = y/pM,M1 = max(M,^/(p«1), y/(pn2))and M2 = min(2M,2y/(pn1),2y/(pn2)). We carry out the summation over m first.Trivially we would take {M2-Mi)q~1 + 0(1) but this is useless because M is goingto be smaller than Q. Therefore we are looking for an explicit formula for the errorterms, with the expectation of obtaining substantial cancellations when summingthem over q. We begin with
M\ <_m ^ M2m — apn\ (modq)
1 = l ^ j - ^ l l + ^ _ i "^-M _ ,/, "'•» - i ^ i I (12)
where \j/(d) = 0 - [ 0 ] - j . To arrive at UP(M, N) we replace (Mj-MJq" 1 , with thehelp of Lemma 3, by
, 2 \<l>(q)J
The first term above contributes to Vp(M, Q) exactly UP(M, Q), while the error term
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE
0{x{q)l4>{q)) contributes
„ . x(q)< 2 N —2— ^ N Q log Q .
Therefore we may write
VP(M,Q) = Up(M, Q)+ Vp(Mt, Q)- Vp{M2, Q) + O(N2Q~1 logQ).
where for L = M1 or M2 we define
145
(13)
Q < q s: 2Qiq,ap) = 1
N/2 < m,n2 < 2JV(mttj.pq) = 1
1/2 < ni/«2 < 2
apni
By Lemma 4 we approximate tp(£) by A(£) with error B(^) giving
* ± 0 AT/2 < n i , n 2 < 2N QqHQ(q,apn\n2) = 1
• (14)
We use v | q°° to mean that each prime that divides v also divides q. Sinceq/<t>(q) = Xv|«=oV^1' w e obtain, by Lemmas 5 and 6 and by partial summation,
i ^e^-^y s / Lpn1-a\ ( q \e h e ah I
Q q Q{q, apn\rt2) —
supPQNJ Q < e, «2g 7
q, apnin2) = 1
l\ an
sup V " 1
(v,dpn\n2) = 1 (r, apn\r\2) = 1
\h\x
Now, summing over n1,n2<2N and h ^ 0 with weight Cfc, (14) withA = MQ~rx 2c yields
146 E. FOUVRY AND H. IWANIEC
Hence relation (13) becomes
Vp(M, Q) = Up(M, Q)
Q-1x5e). (15)
§10. Rearrangement of Wp{M, Q). By definition,
Wp(M,Q)= XQ<q H2Q M < m « 2M (nin2,p) = 1{q, ap) = 1 (m, q) = 1 yjpm < n\, ni ^ 2y/pm
pmni = pmti2 = a{modq)
Here, if (nt, n2) = d > 1 then d > p. Therefore, such terms contribute at most
O{MN2Q'lp-1+MNlogx). (16)
Now, let us consider W*{M, Q)—the contribution to WP(M, Q) of terms with(n1; n2) = 1. Notice that the range for nl, n2 is equal to
<%(q) = {(ni,n2);(n1n2,pq) = l . K , . ^ ) = 1,
JV nnx = n 2 ( m o d q ) , — < n1, n2 < 2N ,j < — < 2 } ,
2 n2
a n d for given q,n1,n2 t h e n u m b e r of m's is given by (12) wi th the same n o t a t i o n forMt and M2. We treat the main term (M2 — Mi)/q as in Section 9. On replacing it by
y iM, < m « M2
we make the total error ^ N2Q~X logQ. Another error of order (16) is made whenrelaxing the condition {nx, n2) = 1 in S(q). The latter operation is necessary toobtain
TP(M,Q)= X J - X X (G q « 1Q r W ) ((mod^) M < m « 2M \ )> < pmn « 2y{q,ap) = 1 (/, q) = 1 (m,q( = 1 pmn = t{modq)
() 1
which we consider as a main term for Wp(M, Q). It is clear, by the above discussion,that we obtain
Wp(M,Q) = Tp(M,Q) + 2Wp(M1,Q)-2Wp(M2,Q)
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE 147
where for L = Mt or M2 we define
WP(L,Q)=Q <q sj 2Q (BI,B2)E[q, ap) = 1 BI > i
§11. Rearrangement of Wp(L,Q). Now approximate i ( ) by /l(^), with error£(£), and expand -4(<!;) and £(£) into Fourier series (see Lemma 4) giving
Wp(L, Q) < AN2\ogN+ f CJW;>fc(L, g) | , (17)h= 1
where
?, ap) = 1 ni > "2
Replace the condition (q,p) = 1 by splitting up the summation over q into p —1arithmetic progressions a(modp), 1 ^ a < p, and detect (q, a) = 1 by the relation
f 1, if (g,a) = 1,
v q \ 0, it (g,a) > 1,
to obtain
wP.k(L'Q)= I ZMvTO(L,g), (18)1 ^ a < p v | a
with
/ P K ) / > 2 ) e ( ^ apn2
g s a(modp) n\ > 112q = 0{modv)
For (nj.njje^q), we reinterpret the condition ny = n2(modq), by writing«i-«2 = qr> so that 1 < r < NQ'\Qr < n^-n2 < 2Qr,
nt—n2 = ar(modpr), n1—n2 = O(modvr) and (n1n2,pr) = 1 .
We may therefore write
1 «r«Af/e(ni,B2)6if(r)
145 E. FOUVRY AND H. IWANIEC
where q = (nl —n2)/r and the range of summation in the inner sum is
N; y < n1?n2 < 2N,n2 < n, < 2n2,
Qr<n1-n2^2Qr,(n1,n2) = l,
{nx n2 •> prv) = 1, nl — n2 = otr(mod pr),
«! = n2 (mod vr) >.
The variables n2 and n1 are of the same order of magnitude. Our intention is to spoilthis 'symmetry' by an appeal to the combinatorial sieve identity. We apply Lemma 1,with some parameter G > p in place of D, giving
Ig < G (n2,prv) = 1 nj = O(mod^i)
+ X IG (02,^^) = 1
say.
+ zG/p !g 9 sc G (12
(20)
§12. Estimate of
obtain, by Lemma 6,
z . Since g = a(modp) and q =. rnj(modn2) we
L-apn2 Lpn2-a q r(Lpn2-a)= 1- a = 1- a
q qpn2 pn2 {nl-n2)pn2 pn2(modi).
Insert this into the inner sum £B1 and remove the factor e(hr(Lpn2 — a)lpn2(nl —n2))
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE
by partial summation, to obtain
149
1=pn2
hxS U P
JV/2 < iVi < 2iVPn2
by Lemma 5. This yields
I Ig ^ G n2
§13. Estimate of £ X
(mod WJ, Lemma 6 yields
(21)
. Since g = a(modp) and q = — rn2
{
Insert this into the inner sum £ n i / a n d remove the factor e(hr(Lpnl —a)/pn1(nl — n2))by partial summation, to obtain
« (i+J1^-) sup yPNQJ N/2 < N, < 2JV JV/2 < n2 < JV 11 < Wl
(^•prv) = 1 MI = O(modg)
an,where c(nx) =/p(9)(n1)e( a^i—-J is independent of n2. By the Cauchy-Schwarz
inequality,
N/2 < n2 < JV ni ^ Ni(n2,prv) = 1 » | s OJmodg)
1/2
n', n " ^ JVi (ri, n) e .^n' = n" = O(modj) (n", n) e J»(r\
(n'n", p) = 1 I
HP-
1/2
However,
np n —ri np
150 E. FOUVRY AND H. IWANIEC
Hence, by Lemma 5,
X e (ahr ( ~ - ^ ]) < (ah{n"-n'), rin"fl2(n', i(n. .(«", n) e J^fr)
Summation over «' and n" yields
n\ n" < 2Nri = n" ~ 0(mod3)
h<h< 2N/g
Gathering all the above results together we finally obtain
I XG/p < g < G n2
(22) 1
§14. Estimate of Wp(L, Q). Collecting (17), (18), (19), (20), (21) and (22) weobtain
Wp(L,Q) ( ^ ^
- 1 x 7 e , (23)
for A = MQ~lx~lt and G = (a ,p)"2 / 5p"1 / 5N2 / 5 .
§15. Estimate of the dispersion Dp(M, Q). If we introduce results (15) and (23)into the definition of DP(M, Q) we obtain
Dp(M, Q) = Xp(M, Q) + RP(M, Q),
where XP(M, Q) stands for the sum of the main terms, i.e.
Xp(M, Q) = Tp(M, Q)-2Up(M, Q)+Up(M, Q)
e < 9 < 2 e M < m « 2 M VV</^ l(modg) \ y < pmn < 2y lT\H> y < pmn « 2y(q,ap) = l ( m , i } | = l (/,()) = 1 pmn = /(mod?) ( n , p ? ) = l
(n,P)= 1
and the error term Rp(M, Q) is
^ ( a , ? ) 1 / 1 0 ? 1 3 ' 1 0 ^ 2 9 / 1 0 ^ - ^ 7 8 . !
ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE 151
It remains to estimate Xp(M, Q). For this, we appeal to the large sieve inequality (2).We first write
Z ( Z / , ( » > - T ^ Z/{modq) \ y < pmn ^ 2y{/, q) = 1 pmn = l(modq)
4>(<i) y < pmn « 2 / y < pmn ^ 2y(n,pq) = 1
Summing this over <j and replacing each x (mod q) by its induced primitive characterX*(modd), d q, we obtain
1
G < « = ; 20 >• < pmn
<Q-2(\ogQ)2
• < 2Q 1 < d < 2Q/e x (mod d) y < pmn ^ 2y(n,ep) = 1
If Q/e ^ (logx)2/1+11 = /J, say, apply the Corollary to Lemma 7, and if Q/e > Happly the large sieve inequality (2), to show that this expression is
whence
and finally
DP(M, Q) 4
Xp(M,Q) <
(24)
§16. Conclusion. By (11),
P <
zo « p < z
152 ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE
on taking D = x[0/2[-i£ for Q sj x11'21 and z ^ x1/883. Hence by (7), (8) and (10) we
obtain (5). This completes the proof of the theorem.
Remark. The limit for the present method turns out to be Q = x10119^'1 in whichcase z ^ xd, with 5 = d(n) a very small positive constant.
References
1. E. Bombieri. "On the large sieve", Mathematika, 12 (1965), 201-225.2. X. Ding and C. D. Pan. "A new mean value theorem", Sri. Sinica, Special Issue (II), (1979), 149-161.3. J. Friedlander and H. Iwaniec. "Quadratic polynomials and quadratic forms", Ada Math., 141 (1978),
1-15.4. P. X. Gallagher. "Bombieri's mean value theorem", Mathematika, 15 (1968), 1-6.5. H. Halberstam and H. -E. Richert. Sieve Methods, (Academic Press, London-New York, 1974).6. C. Hooley. "On the number of divisors of quadratic polynomials", Ada Math., 110 (1963), 97-114.7. C. Hooley. "On the greatest prime factor of a cubic polynomial", J. Reine angew. Math., 303/304
(1978), 21-50.8. H. Iwaniec. "Rosser's sieve", Ada Arith., 36 (1978), 171-202.9. H. D. Kloosterman. "On the representation of numbers in the form ax2 + by1 + cz1 + dt2", Ada Math.,
49 (1926), 407^(64.10. H. L. Montgomery. Topics in Multiplicative Number Theory, Lecture Notes in Math. 227 (Berlin and
New York, 1971).11. Y. Motohashi. "An induction principle for the generalization of Bombieri's Prime Number Theorem",
Proc. Japan Acad., 52 (1976), 273-275.12. H. -E. Richert. Lectures on Sieve Methods, (Tata Inst. of Fund. Research, 1976).13. R. C. Vaughan. "On the estimation of trigonometric sums over primes and related questions", Institut
Mittag-Uffler Report No. 9 (1977).14. A. I. Vinogradov. "The density hypothesis for Dirichlet's L-series" (Russian), Izv. Akad. Nauk SSSR,
Ser. Math. 29 (1965), 903-934; Corrigendum: ibid. 30 (1966), 719-720.15. D. Wolke. "Ober die mittlere Verteilung der Werte zahlentheoretischen Funktionen auf Restklassen I,
Math. Annul, 202 (1973), 1-25.
Dr. E. Fouvry, 10H15: NUMBER THEORY; Multiplicative theory;U.E.R. de Mathematiques et d'Informatique, Distribution of primes and of integers withUniversite de Bordeaux I, specified multiplicative properties.351, Cours de la Liberation,33405 Talence,France.
Dr. H. Iwaniec,U.E.R. de Mathematiques et d'Informatique,Universite de Bordeaux I,351, Cours de la Liberation,33405 Talence,France. Received on the 9th of April, 1980.
Recommended