Prime numbers

1

Prime Numbers

SOLO HERMELIN

Updated: 28.10.12 : 12.09.13

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SOLO

Table of ContentPrimes

Euclid, Euclidean DivisionIntroduction

Prime NumbersEuclid's LemmaFundamental Theorem of ArithmeticPrime Numbers FormulasEuler Zeta Function and the Prime History

Prime Number DistributionPrime Number Theorem (PNT)History of the Asymptotic Law of Distribution of Prime NumbersThe Chebychef Contribution

The Chebyschev Functions (1851)The Chebyschev’s First EstimateThe Chebyschev’s Second Estimate

Riemann's Zeta Function (1859)Riemann Zeta Function ZerosRiemann's Zeta Function PropertiesVon Mangoldt Psi FormulaRiemann's Zeta Function Relations

Abel’s Method of Partial Summation

11 1 1

xdx

xxs

s

ss

s

Möbius Function

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Table of Content (Continue – 1)Primes

The Riemann Prime Number FormulaHadamard Proof of the Prime Number Theorem (1896)Newman’s Proof of the Prime Number Theorem (1980)References End of Presentation

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Table of Content (continue – 2)Primes

AppendicesDefinitionsMellin TransformProof of Riemann's Zeta Function Relations

1Re10

1

zxfordte

tzz

t

tt

z

0

0

1

1sin2

1 i

i

z

de

izz

z

0

0

1

12

sin221

i

i

zz

zz

ide

0

0

1

12

1 i

i

z

dei

zz

zzzzz z

1

2sin22sin2

zzzz zz 112/sin2 1

z

z

z

z zzzz

1

2/12/ 12/12/

Bernoulli Numbers Zeta-Function Values and the Bernoulli Numbers

Zeros of Zeta-Function: ζ (z) = 0

1,1ln1

2ln

12ln

2

1

1

1

2

xasxnnx

xn

n

O

5

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Appendices

11

11

1

1

tispn

sprimep

sn

s

Zeta Function ζ (s) and its Derivative ζ‘ (s)

11

1

tizduuuz

z

zzd

d

z ReRe

ic

ic

z

zdz

x

z

z

ix

'

2

1

1,

1ln

2

tisxdxx

xss

s

primepxp

xprimesofnumberx 1:

Hadamard Product of ζ (s)

Perron’s Formula

Auxiliary Tauberian Theorem

Infinite Series

Series of Functions

Absolute Convergence of Series of Functions

Uniformly Convergence of Sequences and Series

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Appendices

Infinite ProductsThe Mittag-Leffler and Weierstrass Theorems

The Weierstrass Factorization Theorem

The Hadamard Factorization Theorem

Mittag-Leffler’s Expansion Theorem

Generalization of Mittag-Leffler’s Expansion Theorem

Expansion of an Integral Function as an Infinite Product


Hadamard Infinite Product Expansion of Zeta FunctionIntegrationPrime Number Applications

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Introduction

Primes

The start point of this presentation was the book of Marcus de Sautoy , “The Music of the Primes”, 2003, Harper Collins Publisher, which I read during a recreation trip to Crete. The subject was new for me, so to study this topic I turned to the Internet, where I found many related articles. I spend a lot of time trying to partially cover the subject, and this Presentation is the result. It contains no original contributions, but clarifications, in my opinion, of some of the topics.

In order to obtain a coherent presentation and complete some of the proofs more work needs to be done

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SOLO Primes

EuclidEuclid ( Eukleidēs), 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptoleme I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

Euclid" is the anglicized version of the Greek name Εὐκλείδης, meaning "Good Glory".

Euclid of Alexandria Born: about 325 BCDied: about 265 BC

in Alexandria, Egypt

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SOLO Primes

Euclidean Division

In mathematics, and more particularly in arithmetic, the Euclidean division is the usual process of division of integers producing a quotient and a remainder. It can be specified precisely by a theorem stating that these exist uniquely with given properties.

Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that a = bq + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b

Statement of the Theorem

Proof1. Existence

Statue of Euclid in the Oxford University Museum

of Natural History

Consider first the case b < 0. Setting b' = −b and q' = −q, the equation a = bq + r may be rewritten a = b'q' + r and the inequality 0 < r < |b| may be rewritten 0 < r < |b' |. This reduces the existence for the case b < 0 to that of the case b > 0.Similarly, if a < 0 and b > 0, setting a' = −a, q' = −q − 1 and r' = b − r, the equation a = bq + r may be rewritten a' = bq' + r' and the inequality 0 < r < b may be rewritten 0 < r' < b. Thus the proof of the existence is reduced to the case a ≥ 0 and b > 0 and we consider only this case in the remainder of the proof.Let q1 and r1, both nonnegative, such that a = bq1 + r1, for example q1 = 0 and r1 = a. If r1 < b, we are done. Otherwise q2 = q1 + 1 and r2 = r1 − b satisfy a = bq2 + r2 and 0 < r2 < r1. Repeating this process one gets eventually q = qk and r = rk such that a = bq + r and 0 < r < b.This proves the existence and also gives an algorithm to compute the quotient and the remainder. However this algorithm needs q steps and is thus not efficient.

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SOLO Primes

Euclidean Division

In mathematics, and more particularly in arithmetic, the Euclidean division is the usual process of division of integers producing a quotient and a remainder. It can be specified precisely by a theorem stating that these exist uniquely with given properties.

Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that a = bq + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b

Statement of the Theorem

Proof (continue)

2. Uniqueness

Statue of Euclid in the Oxford University Museum

of Natural History

Suppose there exists q, q' , r, r' with 0 ≤ r, r' < |b| such that a = bq + r and a = bq' + r' . Adding the two inequalities 0 ≤ r < |b| and −|b| < −r' ≤ 0 yelds −|b| < r − r' < |b|, that is |r − r' | < |b|.Subtracting the two equations yields: b(q' − q) = (r − r' ). Thus |b| divides |r − r' |. If |r − r' | ≠ 0 this implies |b| < |r − r' |, contradicting previous inequality. Thus, r = r' and b(q' − q) = 0. As b ≠ 0, this implies q = q' , proving uniqueness.

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SOLO Primes

Prime Numbers

Prime Number Definition:A positive integer number p is prime if for all positive integers 1≤ a ≤p, we have for all the Euclidean Divisions

p = a q + rthe reminder r = 0 only for (q=p, a=1) or (q=1, a=p).A Prime Number is divisible only by 1 or by itself.

Proposition 20, Book IX of the Euclide’s Elements: “There are Infinitely many Primes”

Euclid's proof

Consider any finite set S of primes. The key idea is to consider the product of all these numbers plus one:

Sp

pN 1

Like any other natural number, N is divisible by at least one prime number (it is possible that N itself is prime).

None of the primes by which N is divisible can be members of the finite set S of primes with which we started, because dividing N by any of these leaves a remainder of 1. Therefore the primes by which N is divisible are additional primes beyond the ones we started with. Thus any finite set of primes can be extended to a larger finite set of primes.

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Prime Numbers

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67

71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163

167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269

271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383

389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499

503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619

631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751

757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881

883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997

Here is a list of all the prime numbers up to 1,000:

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Run This

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http://en.wikipedia.org/wiki/File:Sieve_of_Eratosthenes_animation.gif

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SOLO Primes Euclid's Lemma

In number theory, Euclid's lemma (also called Euclid's first theorem) is a lemma that captures one of the fundamental properties of prime numbers. It states that if a prime divides the product of two numbers, it must divide at least one of the factors. For example since 133 × 143 = 19019 is divisible by 19, one or both of 133 or 143 must be as well. In fact, 19 × 7 = 133. It is used in the proof of the fundamental theorem of arithmetic.

Let p be a prime number, and assume p divides the product of two integers a and b. Then p divides a or p divides b (or perhaps both).

Divisibility Definition: Assume a ≠ 0 and let b be any integer. If there is an integer q such that b = a.q, a is said to divide b; a is a divisor of b and b is a multiple of a. Notation of a divide b is a|b.

The lemma first appears as proposition 30 in Book VII of Euclid's Elements. It is included in practically every book that covers elementary number theory

Proof:

211221212211

222

111

rrrmrmpmmprpmrpmba

prrpmb

prrpma

Using Euclidean Division Theorem

Since p|a.b we must have r1.r2=0 meaning r1=0, or r2=0, or r1=0 and r2=0.

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SOLO Primes

Fundamental Theorem of ArithmeticIn number theory, the fundamental theorem of arithmetic (also called the unique factorization theorem or the unique-prime-factorization theorem) states (existence) that every integer greater than 1 is either prime itself or is the product of prime numbers, and (uniqueness) that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the fundamental theorem. Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.

Canonical representation of a positive integerEvery positive integer n > 1 can be represented in exactly one way as a product of prime powers:

k

iik

ik ppppn1

2121

Proof of Fundamental Theorem of Arithmetic

Existence

By inspection, each of the small natural numbers 1, 2, 3, 4, ... is the product of primes. This is the basis for a proof by induction. Assume it is true for all numbers less than n. If n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n. By the induction hypothesis, a = p1p2...pn and b = q1q2...qm are products of primes. But then n = ab = p1p2...pnq1q2...qm is the product of primes

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Fundamental Theorem of Arithmetic

Canonical representation of a positive integerEvery positive integer n > 1 can be represented in exactly one way as a product of prime powers:

k

iik

ik ppppn1

2121

Proof of Fundamental Theorem of Arithmetic (continue)

Uniqueness

Assume that s > 1 is the product of prime numbers in two different ways:

nm qqqppps 2121 We must show m = n and that the qj are a rearrangement of the pi.

By Euclid's lemma p1 must divide one of the qj; relabeling the qj if necessary, say that p1 divides q1. But q1 is prime, so its only divisors are itself and 1. Therefore, p1 = q1, so that

nm qqppp

s 22

1

This can be done for all m of the pi, showing that m ≤ n. If there were any qj left over we would have

which is impossible, since the product of numbers greater than 1 cannot equal 1. Therefore m = n and every qj is a pi.

nmm

qqppp

s

121

1

q.e.d.

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SOLO Primes

Marin Mersenne, Marin Mersennus or

le Père Mersenne (1588 –1648)

Mersenne Prime

In mathematics, a Mersenne number, named after Marin Mersenne (a French monk who began the study of these numbers in the early 17th century), is a positive integer that is one less than a power of two: 12 p

pM

Named after Marin Mersenne

Publication year 1636[1]

Author of publication Regius, H.

Number of known terms 47

Conjectured number of terms

Infinite

Subsequence of Mersenne numbers

First terms 3, 7, 31, 127

Largest known term 243112609 − 1

OEIS index A000668

As of October 2009[ref], 47 Mersenne primes are known. The largest known prime number (243,112,609 – 1) is a Mersenne prime.[3] Since 1997, all newly-found Mersenne primes have been discovered by the "Great Internet Mersenne Prime Search" (GIMPS), a distributed computing project on the Internet.

A basic theorem about Mersenne numbers states that in order for Mp to be a Mersenne prime, the exponent p itself must be a prime number. This rules out primality for numbers such as M4 = 24 − 1 = 15: since the exponent 4 = 2×2 is composite, the theorem predicts that 15 is also composite; indeed, 15 = 3×5

While it is true that only Mersenne numbers Mp, where p = 2, 3, 5, … could be prime - and it was believed by early mathematicians that all such numbers were prime[2] - Mp is very rarely prime even for a prime exponent p. The smallest counterexample is the Mersenne number

89x232047121111 M

Prime Numbers Formulas

http://primes.utm.edu/top20/page.php?id=4

http://primes.utm.edu/top20/page.php?id=4

http://en.wikipedia.org/wiki/File:MarinMersenne.jpg

https://www.google.com/imgres?imgurl=http://4.bp.blogspot.com/-G6tWxXAW4ec/TsZRYSaSayI/AAAAAAAAFZQ/wT6N2DEvmFk/s200/A.jpg&imgrefurl=http://www.departamentodematematica.blogspot.com/&h=135&w=96&sz=2&tbnid=48u6VB0Quo__AM&tbnh=0&tbnw=0&zoom=1&usg=__ofQZMgbE1MkQbUXnCCDPBpaaFOE=&docid=MkhhfCWbTW46wM&sa=X&ei=s1gmUKKcA-Sf0QX6s4GwCg&ved=0CIgBENUX

http://en.wikipedia.org/wiki/File:MarinMersenne.jpg

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Goldbach’s Conjecture

Christian Goldbach (1690 –1764)

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture has been shown to be correct[2] up through 4 × 1018 and is generally assumed to be true, but no mathematical proof exists despite considerable effort

History:On 7 June 1742, the German mathematician Christian Goldbach (originally of Brandenburg-Prussia) wrote a letter to Leonhard Euler (letter XLIII)[4] in which he proposed the following conjecture:

Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units

He then proposed a second conjecture in the margin of his letter

Every integer greater than 2 can be written as the sum of three primes

The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time

Prime Numbers Formulas

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http://en.wikipedia.org/wiki/File:Chebyshev.jpg

SOLO Primes

Euler Zeta Function and the Prime History

232 4

1

3

1

2

11

In 1650 Mengoli asked if a solution exists for

P. Mengoli1626 - 1686

The problem was tackled by Wallis, Leibniz, Bernoulli family, without success. The solution was given by the young Euler in 1735. The problem was named “Basel Problem” for Basel the town of Bernoulli and Euler.

Euler started from Taylor series expansion of the sine function

!7!5!3

sin753 xxx

xx

Dividing by x, he obtained

!7!5!3

1sin 642 xxx

x

x

The roots of the left side are x =±π, ±2π, ±3π,…. However sinx/x is not a polynomial, but Euler assumed (and check it by numerical computation) that it can be factorized using its roots as

2

2

2

2

2

2

91

411

21

2111

sin

xxxxxxx

x

x

Leonhard Euler(1707 – 1783)

SOLO Primes

!7!5!3

1sin 642 xxx

x

x

2

2

2

2

2

2

91

411

sin

xxx

x

x

Leonhard Euler(1707 – 1783) If we formally multiply out this product and collect all the x2 terms, we

see that the x2 coefficient of sin(x)/x is

122222

11

9

1

4

11

n n

But from the original infinite series expansion of sin(x)/x, the coefficient of x2 is −1/(3!) = −1/6. These two coefficients must be equal; thus,

1

22

11

6

1

n n 6

1 2

12

n n

Euler extend this to a general function, Euler Zeta Function

,4,3,24

1

3

1

2

11: nn

nnn The sum diverges for n ≤ 1 and

converges for n > 1.

Euler computed the sum for n up to n = 26. Some of the values are given here

,9450

8,945

6,90

4,6

28642

Euler checked the sum for a finite number of terms.

Euler Zeta Function and the Prime History (continue – 1)

SOLO Primes

Euler Product Formula for the Zeta Function

Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series), published by St Petersburg Academy in 1737

primepx

nx pn 1

11

1

where the left hand side equals the Euler Zeta Function

Euler Proof of the Product Formula

xxxxx

s8

1

6

1

4

1

2

1

2

1

xxxxxxxx

13

1

11

1

9

1

7

1

5

1

3

11

2

11

xxxxxxxxx

33

1

27

1

21

1

15

1

9

1

3

1

2

11

3

1

xxxxxxxx

17

1

13

1

11

1

7

1

5

11

2

11

3

11

all elements having a factor of 3 or 2 (or both) are removed

xxxx

nxn

x5

1

4

1

3

1

2

11

1

1

converges for integer x > 1

all elements having a factor of 2 are removed

Leonhard Euler(1707 – 1`783)

EulerZeta Function and the Prime History (continue – 2)

SOLO Primes


Euler Product Formula for the Zeta Function

primepx

nx pn

x1

11

1

Euler Proof of the Product Formula (continue)

xxxxxxxx

17

1

13

1

11

1

7

1

5

11

2

11

3

11

Repeating infinitely, all the non-prime elements are removed, and we get:

12

11

3

11

5

11

7

11

11

11

13

11

17

11

x

xxxxxxx

Dividing both sides by everything but the ζ(s) we obtain

xxxxxx

x

131

1111

171

151

131

121

1

1

Therefore

primepx

nx pn

x1

11

1


http://en.wikipedia.org/wiki/File:Leonhard_Euler_2.jpg

SOLO Primes


Euler Product Formula for the Riemann Zeta Function

primeps

ns pn

s1

11

1

Another Proof:

According to Fundamental Theorem of Arithmetic: Every positive integer n > 1 can be represented by exactly one way as a product of prime powers

integer,21

21 iik primeppppn k

1

211

211

n

s

kn

skppp

ns

primeps

primep k

sk

n

s

kn

s ppppp

ns k

1

11

1121

1

21

Since in the sum n covers all the integers, for each prime there are the powers of al integers k ϵ [1,∞)



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SOLO Primes

The Euler zeta function, ζ(s), is a function is the sum of the infinite series

1

1

nxn

x

Let compute

1,ln

1,1

1

1

1

1px

ps

x

dxx

s

s

According to Maclaurin – Euler Integral Convergence Test for Infinite Seriesthe integral and therefore the series are divergent for p ≤ 1, convergent for p > 1.


Euler Zeta Function and the Prime History (continue – 5)

Euler Zeta Function for x > 1

0823.190

1

2

114

202.11

2

113

645.16

1

2

112

612.22/3

1

2

111

2

10

4

44

33

2

22

n

n

n

n


SOLO Primes

Euler Product Formula

primeps

ns pn

s1

11

1

Another Proof of the Product Formula

Start with the following geometric series expansion

skssss ppppp

11111

1

132

When , we have |p−s| < 1 and this series converges absolutely

Hence we may take a finite number of factors, multiply them together, and rearrange terms. Taking all the primes p up to some prime number limit q, we have

1

1

1

1

qsqps np

s

where σ is the real part of s. By the fundamental theorem of arithmetic, the partial product when expanded out gives a sum consisting of those terms n−s where n is a product of primes less than or equal to q. The inequality results from the fact that therefore only integers larger than q can fail to appear in this expanded out partial product. Since the difference between the partial product and ζ(s) goes to zero when σ > 1, we have convergence in this region.



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In number theory, the Prime Number Theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers.

Prime Number Distribution

SOLO Primes

Since a general formula for the Prime determination couldn’t be found, the attention was driven to the following question: How to find a function that defines the number of primes less or equal to a given number x? This function was named π (x)

primepxp

xprimesofnumberx 1:

The first question that was unsuccessful tackled was: Given a integer number N, how to find the Prime Number P, less then N, and as closed as possible to N.

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27

In number theory, the Prime Number Theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers.

Prime Number Theorem (PNT) Let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The Prime Number Theorem then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1, which is expressed by the formula

Prime Number Theorem (PNT)

1

ln/lim

xx

xx

π(x)

x / ln(x)

SOLO Primes

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http://en.wikipedia.org/wiki/File:PrimeNumberTheorem.svg

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SOLO Primes

History of the Asymptotic Law of Distribution of Prime Numbers

Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function a/(A ln(a) + B), where A and B are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture, with A = 1 and B = −1.08366.

Adrien-Marie Legendre )1752 – 1833 (

Carl Friedrich Gauss considered the same question: "Ins Jahr 1792 oder 1793", according to his own recollection nearly sixty years later in a letter to Encke (1849), he wrote in his logarithm table (he was then 15 or 16) the short note "Primzahlen unter

But Gauss never published this conjecture.

BaA

aa

ln

a

aa

ln

29

SOLO Primes


Later Gauss came up with a new approximating function, the logarithmic integral Li (x)

x

u

duxLi

2 ln:

Calculating 1000

1000

xxx

Computing by hand, it seams that Δ(x) tends to zero ,but very slowly. To see how slow computing the inverse of Δ(x) it was found that

xx ln/1

Meaning that x

xln

1

Define

Carl Friedrich Gauss(1777 – 1855)

xLi

x

x

x

ln

30

x π(x) π(x) − x / ln x π(x) / (x / ln x) li(x) − π(x) x / π(x)10 4 −0.3 0.921 2.2 2.500

102 25 3.3 1.151 5.1 4.000103 168 23 1.161 10 5.952104 1,229 143 1.132 17 8.137105 9,592 906 1.104 38 10.425106 78,498 6,116 1.084 130 12.740107 664,579 44,158 1.071 339 15.047108 5,761,455 332,774 1.061 754 17.357109 50,847,534 2,592,592 1.054 1,701 19.667

1010 455,052,511 20,758,029 1.048 3,104 21.9751011 4,118,054,813 169,923,159 1.043 11,588 24.2831012 37,607,912,018 1,416,705,193 1.039 38,263 26.5901013 346,065,536,839 11,992,858,452 1.034 108,971 28.896

1014 3,204,941,750,802 102,838,308,636 1.033 314,890 31.202

1015 29,844,570,422,669 891,604,962,452 1.031 1,052,619 33.507

1016 279,238,341,033,925 7,804,289,844,393 1.029 3,214,632 35.812

1017 2,623,557,157,654,233 68,883,734,693,281 1.027 7,956,589 38.116

1018 24,739,954,287,740,860 612,483,070,893,536 1.025 21,949,555 40.420

1019 234,057,667,276,344,607 5,481,624,169,369,960 1.024 99,877,775 42.725

1020 2,220,819,602,560,918,840 49,347,193,044,659,701 1.023 222,744,644 45.028

1021 21,127,269,486,018,731,928 446,579,871,578,168,707 1.022 597,394,254 47.332

1022 201,467,286,689,315,906,290 4,060,704,006,019,620,994 1.021 1,932,355,208 49.636

1023 1,925,320,391,606,803,968,923 37,083,513,766,578,631,309 1.020 7,250,186,216 51.939

SOLO Primes


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Both Gauss's formulas imply the same conjectured asymptotic equivalence of π(x) , x / lnx and Li (x) stated above, although it turned out that Gauss's Li (x) approximation is considerably better if one considers the differences instead of quotients. By using L’Hopital theorem we can see that

11ln

lnlim

ln

1lnln1

lim

ln

limln/

lim

2

x

x

x

xx

xx

xdd

xLixd

d

xx

xLixxxx

Example:

6115ln/,128,78498,106 nnnnnLinn


32

SOLO Primes

Gauss's function compared to the true number of primes

Gauss's guess was based on throwing a dice with one side marked "prime" and the others all blank. The number of sides on the dice increases as we test larger numbers and Gauss discovered that the logarithm function could tell him the number of sides needed. For example, to test primes around 1,000 requires a six-sided dice. To make his guess at the number of primes, Gauss assumed that a six-sided dice would land exactly one in six times on the prime side. But of course it is very unlikely that a dice thrown 6,000 times will land exactly 1,000 times on the prime side. A fair dice is allowed to over- or under-estimate this score. But was there any way to understand how to get from Gauss's theoretical guess to the way the prime number dice had really landed? Aged 33, Riemann, now working in Göttingen, discovered that music could explain how to change Gauss's graph into the staircase graph that really counted the primes.


University of Göttingen


33

SOLO Primes

John Edensor Littlewood1885 - 1977

xx

xxLix

lnlnln

ln2/1

.10341010x

.1031010x

Gauss asserted that π (x) < Li (x). Toward the end of his 1859 paper Riemann makes the same assertion. Using computation this was proved to be true for all x < 108.In 1914 Litlewood showed that π (x) – Li (x) changes sign infinitely often. He showed that there is a constant K > 0 such that

is greater than K for arbitrarily large x and less than –K for arbitrarily large x.Litlewood’s method helped Skewes, who in 1933, showed that there is at least one sign change at x for some

Skewes proof required the Riemann Hypothesis. In 1955 he obtained a bound without using the Riemann Hypothesis. This new bound was

Skewes large bound can be reduced substantially. In 1966 Sherman Leham showed that between 1.53x101165 and 1.65x101165 there are more than 10500 successive integers x for which π (x) > Li (x). Lehman work suggest there is no sign change before 1020.In 1987 Riele showed that between 6.62x10370 and 6.69x10370 there are more than 10180

successive integers for which π (x) > Li (x).


34

SOLO Primes


In 1837 Johann Peter Gustav Lejeune Dirichlet introduced Dirichlet Series

Johann Peter Gustav Lejeune Dirichlet

)1805– 1859 (

1

:ˆn

sn

nfsf

is convergent for Re (s) > c if f (n) = O (n c-1 ) as n → ∞.

Given the Perron’s Formula

Oskar Perron ( 1880 – 1975)

011

10

2

1

xif

xifds

s

x

i

i

i

n

then

xnn

i

i ns

si

i

s nfnxif

nxifnf

s

ds

n

nfx

is

dssfx

i 111 1

0

2

1ˆ2

1

For f (n) = 1 we obtain the Zeta Function

1

1:

nsn

s

therefore

xn

i

i

s

s

dssx

i 1

12

1

SOLO Primes


In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s) (for real values of the argument "s", as are works of Leonhard Euler, as early as 1737) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity exists at all, then it is necessarily equal to one.[2] He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all x.[3]


Joseph Louis François Bertrand

(1822 –1900)

Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2.

5/6,30/532log 1230/15/13/12/1

1 ccc where , and N is sufficiently large.

N

NcN

N

Nc

lnln 11

Pafnuty Lvovich Chebyshev

) ) 1821 – 1894

http://en.wikipedia.org/wiki/File:Bertrand.jpg



http://en.wikipedia.org/wiki/File:Bertrand.jpg

SOLO Primes


Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject. Riemann introduced revolutionary ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending these deep ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + i t with t > 0

Georg Friedrich Bernhard Riemann

)1826– 1866(

Jacques Salomon Hadamard

(1865 –1963)

Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin

(1866 1962)

SOLO Primes


During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős (1949). While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, and later proofs have introduced various simplifications through the use of Tauberian theorems but remained difficult to digest, a surprisingly short proof was discovered in 1980 by American mathematician Donald J. Newman. Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis

Atle Selberg (1917 –2007)

Paul Erdős (1913 –1996)

Donald J. Newman ( 1930 –2007)

Return to TOC

38

SOLO Primes The Chebychef Contribution

integeres,1

2121

ii

m

i

ki

km

kk kprimespppppn im

The starting point is that any positive number can be factored into a unit product of powers of distinct primes

integeres,lnlnlnlnln1

2211

ii

m

iiimm kprimesppkpkpkpkn

The utility of this formula is enhanced by the use of von Mangold symbol Λ (n)

otherwise

kandpprimesomeforpnifpn

k

0

1integerln

Hans Carl Friederich von Mangold (1854 – 1925)

The symbol Σj|n will be used to denote a sum on j where j runs through all of the positive divisors of the positive integer n. With this notation we have:

m

iii

nj

pkjn1|

lnln

To prove this note that from and the definition of Λ (j) the only nonzero terms that can appear on the right side are ln p1,ln p2,…,ln pk. Moreover p1 appears for j=p1, j=p1

2,…,j=p1k1. Thus ln p1 appears exactly

k1 times. Similarly p2 appears exactly k2 times, etc. q.e.d.

mkm

kk pppn 2121

Since we have products a most useful formula is obtained by using natural logarithm


) ) 1821 – 1894

Return to TOC

39

SOLO Primes

otherwise


k

0

1integerln Von Mangoldt Function1895


) ) 1821 – 1894

The Chebyschev Functions (1851)

primepxp

px ln:Chebyschev Theta Function

primepxpxn k

pnx ln:Chebyschev Psi Function

From the definition of Chebyschev Psi Function and of Λ (j)

3/12/1

lnlnlnln:32

xxx

ppppnx

primepxp

primepxp

primepxp

primepxpxn k

http://en.wikipedia.org/wiki/File:Chebyshev.jpg

40

SOLO Primes

3ln2ln7ln5ln2ln3ln2lnln primep

xpk

px

otherwise


k

0

1integerln

The Chebyschev Functions (continue - 1)

7ln5ln3ln2lnln:1010

primep

p

px

Example: x = 10

Prime Numbers p < x = 10 :p: 2, 3, 5, 7

Prime Numbers p2 < x = 10 :p2: 22=4, 32=9

Prime Numbers p3 < x = 10 :p3: 23=8,

,010,3ln9,2ln8,7ln7,06

,5ln5,2ln4,3ln3,2ln2,01

7ln5ln3ln22ln3:1010

xn

nx

1621028 43 x

2ln

10ln32ln410ln2ln3

10..integral: xxtsx

primepxp

primepxp

pp

xpx

k

lnln

lnln

41

SOLO Primes

3/2/ln: xxxpnx

primepxpxn k

otherwise


k

0



primepxp

px ln:

Theorem x

x

x

x

xx

xxxx

limlimln/

lim

Proof:

primepxp

xprimesofnumberx 1:

primepxp

xp

primepxp

xpxxxxxk

1lnln3/2/lnln

Define:

11ln

ln/::,: 321

xx

x

xx

xL

x

xL

x

xL

primepxp

Therefore: 321 LLL

One the other hand, if 0 < α <1, x > 1, then: x > α → ln x > ln α

xxxxxxxxppx

xx

xpxxpx

xp

xpxpx

primepxp

lnlnln1lnlnln:10lnln

Return to Newman Proof of PNT

Chebyshev didn’t prove that the limit is 1.

42

SOLO Primes

3/2/ln: xxxpnx

primepxpxn k


primepxp

px ln:

Theorem x

x

x

x

xx

xxxx

limlimln/

lim

Proof (continue):

primepxp

xprimesofnumberx 1:

Define:

11ln

ln/::,: 321

xx

x

xx

xL

x

xL

x

xL

primepxp

321 LLL

xxxx ln

Dividing the inequality by x > 1 we obtain:

1

lnln

x

x

x

xx

x

x

Keep α fixed and x → ∞ we obtain: 0ln

lim10

1

x

xx

Hence:

31 limln

limlimlim Lx

xxL

x

xxxxx

gives: 321 limlimlim LLLxxx

q.e.d.

Tacking α→1: 31 limlim LLxx

together with 321 LLL

Return to TOC

xx OReturn to

43

SOLO Primes

xx O

The Chebyschev’s First Estimate

primepxp

px ln:

Theorem

Proof: Start with the Binomial formula

121

1212222112

integer

2

0

22

nn

nnnn

n

n

k

nn

k

nn

nnpppp

npn

eeeepn

nnpnpnpnnpn

2lnlnlnln

2

2222

Taking natural algorithm from both sides, we obtain nnn 22ln2

Definition of O:We say that f (x) = O (g (x)) if exists a constant k > 0 such that |f (x)| < k |g (x)|

nk

primepnpnnpnnkn

kbbydividednotispcpknnnna1222

:&:12122

cb

apkcbka

npnnknkn

212

:

44

SOLO Primes

xx O

The Chebyschev’s First Estimate

primepxp

px ln:

Theorem

Proof (continue):

q.e.d.


Let be r the minimal integer such that 2r > x. Then

xxx 2ln12/

rrrr

xxxxxxxx

xxx

22222222 1122

Therefore xx O

xxxxx rr

jj

r

jjj

2ln12

21

1

21

12ln1

22ln1

22

1

0

1

01

Taking natural algorithm from both sides, we obtain nnn 22ln2Define [x] the biggest integer less than x; i.e. 0 < x – [x] < 1 Then

xxx

xxxx

xxxx 2ln12ln

22ln

222

22

22/

Return to TOC

45

SOLO Primes

xx O

The Chebyschev’s Second Estimate

primepxp

px ln:

Theorem

Proof:


3/2/ln: xxxpnx

primepxpxn k

For 0 < δ < 1 and y = x1-δ, we have

x

xx

y

xyyx

y

xp

yp

yyy

primepxpy

primepxp

primepxpy

primepxpy

primepyp

ln1

1

ln1

lnln

ln

1ln

ln

11&1

1

Therefore

x

x

x

x

x

x

x

x

xx

x

x

x

x

x

1

1ln

1

1ln

ln/

We also proved that xx

x

x

x

x

x

ln/

46

SOLO Primes

xx O

The Chebyschev’s Second Estimate

primepxp

px ln:

Theorem

Proof (continue):


3/2/ln: xxxpnx

primepxpxn k

For x → ∞ and δ→0 we have

For 0 < δ < 1 and y = x1-δ, we have

2ln12

1

1ln

1

1ln 2ln12

x

x

x

x

x

x

x

x xx

0ln

x

x

xxx

2ln12

Therefore xx Oq.e.d.

Return to TOC

47

SOLO Primes

Riemann's Zeta Function (1859)The Riemann Zeta Function or Euler–Riemann Zeta Function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series

tisn

sn

s

1

1

“On the Number of Primes Less Than a Given Magnitude”, 7 page paper offered to the Monatsberichte der Berliner Akademie on October 19, 1859. The exact publication date is unknown.

ssss ss

1

2sin12 1

where Γ(s) is the Gamma Function, which is an equality of Meromorphic Functions valid on the whole complex plane. This equation relates values of the Riemann Zeta Function at the points s and 1 − s. The functional equation (owing to the properties of sin ) implies that ζ(s) has a simple zero at each even negative integer s = −2n — these are known as the trivial zeros of ζ(s). For s an even positive integer, the product sin(πs/2)Γ(1−s) is Regular and the functional equation relates the values of the Riemann Zeta Function at odd negative integers and even positive integers.

Georg Friedrich Bernhard Riemann )1826– 1866(

Return to TOC

To construct the analytic Continuation of the Zeta Function, Riemann established the relation (see proof ).

Graph showing the Trivial Zeros, the Critical Strip and the Critical Line of ζ (s) zeros.

SOLO Primes

,2,11

1 1

nn

Bn nn

Those roots are called the Trivial Zeros of the Zeta Function. The remaining zeros of ζ (s) are called Nontrivial Zeros or Critical Roots of the Zeta Function.The Nontrivial Zeros are located on a Critical Strip defined by 0 < σ < 1.

Since Bn+1 = 0 for n + 1 odd (n even) we also have ,2,102 mm

We found

tispn

sprimep

zn

sRe

1

11

1

Riemann Zeta Function Zeros

Since the product contains no zero factors we see that ζ (z) ≠ 0 for Re {z} >1.

Riemann Conjecture in his paper was that all Zeta Function Nontrivial Zeros are located at σ = ½. This Conjecture was not proved and is named One of the Greatest Mysteries in Mathematics.

Bn are the Bernoulli numbers

49

SOLO Primes

Riemann's Zeta Function

Specific Values

,3,2,1,0

!22

212

221 n

n

Bn

nnn

For any positive even number 2n

where B2n are the Bernoulli numbers.

,3,2,11

1 1

nn

Bn nnFor negative integers one has

Therefore ζ vanishes at the negative even integers ζ (-2m) = 0 since B2m+1 = 0 for all m , m=1,2,…

,3,2,12

121 2 mB

mm m

It is easy to show that the last equation is equivalent with

2

1

210

101 0

BB

50

SOLO Primes


The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series

tisn

sn

s

1

1

which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

Georg Friedrich BernhardRiemann

1826 - 1866

51

SOLO Primes

Riemann's Zeta Function tisn

sn

s

1

1

Georg Friedrich Bernhard Riemann )1826– 1866(

Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): colors close to black denote values close to zero, while hue encodes the value's argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(s) = 1/2 are its zeros. Values with arguments close to zero including positive reals on the real half-line are presented in red

52

SOLO Primes

Riemann imaginary landscape

Graph showing the Trivial Zeros, the Critical Strip and the Gritical Line of ζ (s) zeros.

Modulus |ζ s)| ploted over the complex plane


53

SOLO Primes

The plots above show the real and imaginary parts of plotted in the complex plane together with the complex modulus of ζ (s) . As can be seen, in right half-plane, the function is fairly flat, but with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros of ζ (s) lie.


54

Riemann's Zeta FunctionPrimes

55

Re ζ (s) in the original domain, Re s > 1.

Re ζ (s) after Riemann’s extension.


Primes

56

SOLO Primes

The position of the complex zeros can be seen slightly more easily by plotting the contours of zero real (red) and imaginary (blue) parts, as illustrated above. The zeros (indicated as black dots) occur where the curves intersect

The figures bellow highlight the zeros in the complex plane by plotting |ζ(s)|) where the zeros are dips) and 1/|ζ(s)) where the zeros are peaks).


57

The Riemann Hypothesis

The Non-Trivial Zeros ρ of ζ (s) has Re ρ = 1/2


Primes

58

SOLO Primes

Year Number of zeros Computed by1859 (approx.) 1 B. Riemann1903 15 J. P. Gram1914 79 R. J. Backlund1925 138 J. I. Hutchinson1935 1,041 E. C. Titchmarsh1953 1,104 A. M. Turing1956 15,000 D. H. Lehmer1956 25,000 D. H. Lehmer1958 35,337 N. A. Meller1966 250,000 R. S. Lehman1968 3,500,000 J. B. Rosser, et al.1977 40,000,000 R. P. Brent1979 81,000,001 R. P. Brent1982 200,000,001 R. P. Brent, et al.1983 300,000,001 J. van de Lune, H. J. J. te Riele1986 1,500,000,001 J. van de Lune, et al.2001 10,000,000,000 J. van de Lune (unpublished)2004 900,000,000,000 S. Wedeniwski2004 10,000,000,000,000 X. Gourdon

Computation of the Non-trivial Zeros of the Riemann Zeta Function.All were on the Critical Line σ = ½.

Riemann's Zeta FunctionRiemann Conjecture in his paper was that all Zeta Function Nontrivial Zeros are located at σ = ½. This Conjecture was not proved and is named One of the Greatest Mysteries in Mathematics.

Return to TOC

59

SOLO Primes Riemann's Zeta Function Properties

1

1'uduu

ss

s s

We found

ic

ic

s

sds

x

s

s

ix

'

2

1Mellin Transform

1,

1ln

2

tisxdxx

xss

s

primepxp

xprimesofnumberx 1:

100

2/12ln

12/112

Re

s

es

ss

es

Hadamard

γ is the Euler-Mascheroni constant

γ=0.57721566490153286060651

2210 11

1

1ss

ss

1

lnlnlim

!

1 1

k

N

m

m

k

k

Nm

k

N

k

k

3/2/ln: xxxpnx

primepxpxn k

otherwise


k

0


primepxp

px ln:

60


We found

11

11

1

xdxxxss

s s

11

1lim1 1

11

s

ss

s

ss

sss ResRes

primesdistinctkofproducttheisnif

factorprimemultiplesomecontainsnif

nif

n

FunctionbiusoM

k1

0

11

10

11

| nif

nifd

nd

1

1

nsn

n

s

11

1 xdxxss s

Mellin Transform

ic

ic

s sds

sx

ix

2

1

61


We found

1

1

4

1

3

1

2

1

1

/1

1

4

1

3

1

2

1:

1

0

11

n

n

k

n

n

xn

xxxxxJ

FunctionbiusoM



nif

n

xJn

nx

1,

1

ln

2

tisxd

xx

x

s

ss

primepxp

xprimesofnumberx 1:

1

11

n

nxn

xJ

ic

ic

s

s

ss

n

n

sds

sx

ixJ

xdxxJs

s

xdxxJxdxxJs

s

Jxxxn

xxxxxJ

ln

2

1

ln

ln

000101

4

1

3

1

2

1:

0

1

0

1

1

1

1

1

4

1

3

1

2

1

p

sssxdxdv

xu

s pxdxxxxdxxss

10

11

11

62

SOLO Primes Riemann's Zeta Function

We found

x

n

n

x

dxxLi

xLin

nxR

0

1

/1

ln:

:

1 !

lnlnln

n

n

nn

xxxLi

1

1

11 1

1 1!1

!1

1

m

mmn

n

nnm n

m

m

mmm

t

n

n

mm

txR

nm

63

SOLO Primes

xxLix ln O

x

t

tdxLi

2 ln:

primepxp

xprimesofnumberx 1:

xxx 2ln O

nnLipn2/51 lnO

Constantln

11 2/1 Eulerx

x

e

pxp

O

0Im0

2/12ln

12/112

s

es

ss

es Hadamard

Definition of O:We say that f (x) = O (g (x)) if exists a constant k > 0 such that

|f (x)| < k |g (x)| Return to TOC

64

SOLO Primes

Riemann Zeta Function

11

10

2

1

2:Re aif

aifds

s

a

i ss

s

Special case of Perron’s Formula

Chebychev Psi Function

2:Re

11

2:Re/

1

ln

2

1/ln

2

1ln:

ss

s

mprimep

xpms

mprimep

xp ss

smPerron

xa

mprimep

xp

dss

x

p

p

ids

s

pxp

ipx

mmm

m

We were able to swap the infinite sum and the infinite integral since the terms are convergent as Re (s) = 2

11

1lnln1

tispm

psprimep m

ms

SeriesTaylor

primep

s

1

ln

1

ln1'ln

11

1

tis

p

p

p

pp

s

ss

sd

d

primep primep mms

Taylor

p

s

s

s

tispn

sprimep

sn

s

1

1

11

1

2:Re2:Re

1

'

2

1ln

2

1

ss

s

ss

s

mprimep

xpms

dss

x

s

s

ids

s

x

p

p

ix

m

Von Mangoldt Psi FormulaHans Carl Friederich von Mangoldt 1895

2/12

0Re0

1

1ln0

0'ln:

xx

xpx

mprimep

xpm

65

SOLO Primes

2:Re2:Re

1

'

2

1ln

2

1

ss

s

ss

s

mprimep

xpms

dss

x

s

s

ids

s

x

p

p

ix

m

Von Mangoldt Psi Formula(continue – 1)

1 42

t

10'

xdss

x

s

s

LC

s

LC

R

eRsCj

L

0cos2

2:

R

Critical Region(non-trivial zeros)

ζ (s) = 0trivial zerosζ (s) = 0

26 4

Therefore

Define a semi-circular path CL (left side), with s=2 as the origin., and R → ∞.

0''

sincos

''

0cos0

coscos

sincos

,,

R

xC

R

C

R

C

iiRR

C

s

LL

RLRL

dxs

sdR

R

x

s

s

deRiiRR

x

s

sds

s

x

s

s

LL Cs

s

C

s

ss

s

ss

s

dss

x

s

s

ids

s

x

s

s

ids

s

x

s

s

ids

s

x

s

s

ix

2Re

0

2:Re2:Re

'

2

1'

2

1'

2

1'

2

1

s

x

s

s

s

x

s

s

s

x

s

s

s

x

s

s s

sofzeros

s

s

s

s

s

Cs L

'Residues

'Residue

'Residue

'Residues

102)Re(

2/12

0Re0

1

1ln0

0'ln:

xx

xpx

mprimep

xpm

66

SOLO Primes


LCs

s

dss

x

s

s

ix

2Re

'

2

1

1Re0

00

,...4,2

00

1

1

0

0

10

'lim

'lim

1

'1lim

'lim

'Residues

'Residue

'Residue

ZerosTrivialNon

s

ZerosTrivial

sss

s

sofzeros

s

s

s

s

x

s

ss

x

s

ss

x

s

ss

s

x

s

ss

s

x

s

s

s

x

s

s

s

x

s

s

2

10&2ln

2

10'2ln

0

0''lim

0

0

s

x

s

ss

s

1 42

t

10'

xdss

x

s

s

LC

s

LC

R

eRsCj

L

0cos2

2:

R



26 4

2/12

0Re0

1

1ln0

0'ln:

xx

xpx

mprimep

xpm

xx

s

sx

s

ss

ss

1

1

1

1

11'

1lim

1

'1lim

Now we have:

67

SOLO Primes


LCs

s

dss

x

s

s

ix

2Re

'

2

1

1Re0

00

,...4,2

00

1

1

0

0

'lim

'lim

1

'1lim

'lim

ZerosTrivialNon

s

ZerosTrivial

sss

x

s

ss

x

s

ss

x

s

ss

s

x

s

ss

2/12

1

2

1

2

1

00

2

,4,2

00

1ln22

2'2

lim'

lim

x

n

x

n

xn

s

nsx

s

ss

Taylor

n

n

n

n

ns

ZerosTrivial

s

00

00

10

0

'lim

'lim

xx

s

ss

x

s

ss

s

ZerosTrivialNon

s

1 42

t

10'

xdss

x

s

s

LC

s

LC

R

eRsCj

L

0cos2

2:

R



26 4

2/12

0Re0

1

1ln0

0'ln:

xx

xpx

mprimep

xpm

q.e.d.

1'

'

1'lim

'

0

HopitalL

s s

s

We also have:

68

SOLO Primes

Von Mangoldt showed that ψ can also be determined from the non-trivial zeros ρ of the Zeta Function ζ (ρ) = 0

2

2/1lnln

2

010

1

xxxpx

mxp

primepm

Because the zeros ρ are complex, the values xρ/ρ are also complex.But since the nontrivial zeros come in complex-conjugate pairs ρ and ρ*. The values xρ/ρ and xρ*/ρ* are also complex conjugate so all imaginary parts cancel in the infinite sum.

The function xρ/ρ maps the positive reals onto a logarithmic spiral in the complex plane. xρ/ρ and xρ*/ρ* produce complex conjugate spirals (mutual reflections across the real axis. xρ/ρ + xρ*/ρ* =2 Re [xρ/ρ] is a real valued function, a sort of logarithmically – rescaled sinusoid with increased amplitude as pictured bellow:

...)13.14(2/1 i ...)58.37(2/1 i

Von Mangoldt Psi Formula (continue – 4)

69

SOLO Primes


2

2/1lnln

2

010

1

xxxpx

mxp

primepm

Comparing ψ (x) with its approximation via summing the first 50 zeros of the Zeta function.

The Chebyshev Psi Function can be reconstructed by starting with the function x – ln (2π)-1/2 ln (1-1/x2), and then successively adding “spiral wave” functions.


70

SOLO Primes


2

2/1lnln

2

010

1

xxxpx

mxp

primepm

Comparing ψ (x) in the interval x ϵ (2.5, 5.5) with its approximation via summing the first 100 zeros of the Zeta function.

Comparing ψ (x) in the interval x ϵ (2.5, 5.5) with its approximation via summing the first 500 zeros of the Zeta function.

The Chebyshev Psi Function can be reconstructed by starting with the function x – ln (2π)-1/2 ln (1-1/x2), and then successively adding “spiral wave” functions.

2

2/1lnln

2

010

1

xxxpx

mxp

primepm


71

SOLO Primes


2

2/1lnln

2

010

1

xxxpx

mxp

primepm

Let take the derivative of the staircase function ψ (x)

2/12

11'

2

010

1

x

xxxx

xd

d

Since ψ (x) is a staircase function that jumps at each prime power pk, ψ’(x) should be zero except for spikes at

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19,…In the sum

each conjugate pair contributes a waveform (harmonic mode)

21'

2

010

1

x

xxx

, xxeexxx xixi ln1cos2 1ln1ln1111 ImReImImRe

Since 0 < Re ρ < 1, we have -1 < Re (ρ-1) < 0, therefore the amplitude of the waveform is a monotonic decreasing function of x. The frequency of the waveform is related to Im (ρ – 1) ln x is a monotonic increasing function of x.

12 Rex


72

SOLO Primes

The effect of Riemann's harmonics

Riemann's harmonics


73

SOLO Primes


For example here are plots of ψ’(x) using Nρ=10, 50 and 200 pairs of zeros

ψ’(x) is zero except for spikes at2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19,…

Nρ = 10

Nρ = 50

Nρ = 200

2/12

11'

2

010

1

x

xxxx

xd

d

74

SOLO Primes

Each conjugate pair contributes a waveform (harmonic mode) , xxxx ln1cos2 111 ImRe

If the Riemann Hypothesis (R.H. = Re ρ = ½) is true all the harmonics will have the same amplitude xx /22 2/1

If the Riemann Hypothesis is not , that at least one harmonics has a different amplitude then others.


75

SOLO Primes

01

lim

010

x

xx

But independent if the assumption that Riemann Hypothesis is true or false, since we have 0 < Re ρ < 1 for all ρ, we have

From the Explicit Formula for ψ (x)

2

2/1ln

111

2

010

x

x

x

xx

x

Also 02

2/1ln

1lim

2

x

xx

Therefore that proves the Prime Number Theorem.

1lim x

xx


Return to TOC

SOLO Primes

1

1

4

1

3

1

2

1

1

1

1

4

1

3

1

2

1:

ln

n

n

s

xn

xxxxxJ

xdxxJs

s

This sum is only formally infinite, since , as soon as decreases bellow 2, which will happen as soon as n > lnx/ln2. f (x) has jumps of 1/r when x passes a prime power pr. (when x passes a prime p, this is regarded as the prime power p1.)

0/1 nx nx /1

Proof:

primepxp

s

primepxp

s

primepxp

sa

SeriesTaylor

primepxp

s

primepxp

s

ppp

pps

321ln

1

3

1

2

1

1ln1lnln

Riemann's Zeta Function Relations

SOLO Primes

1

1

4

1

3

1

2

1

1

1

1

4

1

3

1

2

1:

ln

n

n

s

xn

xxxxxJ

xdxxJs

s

Proof (continue – 1):

primepxp

s

primepxp

s

primepxp

s ppps 32

3

1

2

1ln

Using Stieltjes’ Integrals and performing Integration by Parts, we obtain

p

sssxdxdv

xu

s pxdxxxxdxxss

10

11

11

This follows since and d π (x) will increase by 1 when x is a prime number p, and will be zero between primes.

0lim000

0

xxx s

x

In the same way

p

snnsnsxdxdv

xu

sn pxdxxxxdxxss

n

1

1

0

1

1

1

11 1

1


SOLO Primes


1

1

4

1

3

1

2

1

1

1

1

4

1

3

1

2

1:

ln

n

n

s

xn

xxxxxJ

xdxxJs

s


1

1

1

12

1

1

1

32

2

1

3

1

2

1ln

xdxxJs

xdxxsxdxxs

ppps

s

ss

primepxp

s

primepxp

s

primepxp

s

1

11:

n

nxn

xJ

SOLO Primes

ic

ic

s

s

ss

n

n

sds

sx

ixJ

xdxxJs

s

xdxxJxdxxJs

s

Jxxxn

xxxxxJ

ln

2

1

ln

ln

000101

4

1

3

1

2

1:

0

1

0

1

1

1

1

1

4

1

3

1

2

1

1,

1

ln

2

tisxd

xx

x

s

ss

We found the following expressions for ln ζ(s)/s:

0

1 xdxfxsFxf sMM

ic

ic

s sdsFxi

x M1- fsfM

2

1


Return to TOC

SOLO Primes

Abel’s Method of Partial Summation:

1

2 11

1

11 11

N

n

N

nnN

n

iin

N

n

n

iin

N

nnn babababa

N

n

N

nnN

n

iin

N

n

n

iin bababa

1 11

11

1 1

N

n

n

iinn

N

nnN baaba

1 11

11

n

iin

N

nnnnNN

N

nnn bBBaaBaba

1111

1

:

1

n

n

Niels Henrik Abel ( 1802 – 1829)

http://en.wikipedia.org/wiki/File:Niels_Henrik_Abel.jpg

SOLO Primes

Use Abel’s Method of Partial Summation:

n

iin

N

nnnnNN

N

nnn bBBaaBaba

1111

1

:

1lim1

N

n

s

Nnsfor:

by choosing an = n-s, bn = 1, therefore Bn = n

110

1

11lim1limlimn

ssN

n

ss

N

s

N

N

n

s

NnnnnnnNNns

xdxxsxdxns s

nx

n

n

n

s

1

1

1

1 1

Where [x] is the integer, less then x and closer to x

10s.t.integer xxx

11

1 xdxxss s

1

1 1 1

1

1

1 11 11

1

xdx

xxs

s

xsxd

x

xxsxd

x

xsxdxxss

s

s

sss

11 1

1

xdxxxs

s

ss s


Return to TOC

SOLO Primes

1

1 1 1

xdx

xxs

s

ss

s

11

100lim

11 xd

xdxd

x

xdns

s

N

n

s

N

Proof:

Integrating by parts:

11

11

1

1

11

11

11

0

1

,

,1

11

1

ss

s

ssss

xddvxu

xvdxxsdus

x

xdxxs

s

s

x

xdxxs

s

xs

x

dxxxs

x

dxxs

x

dxxs

x

x

x

xds

s

s

10s.t.integer xxx

11

11 1

forconvergesn

xdx

xx

nss

We can see that

We have an Analytic Continuation for by removing the singularity at s = 1of ζ (s). We can see that ζ (s) can a simple pole at s=1, and

1

s

ss

11

1lim1 1

11

s

ss

s

ss

sss ResRes


SOLO Primes

0

1 xdxfxsFxf sMM

ic

ic

s sdsFxi

x M1- fsfM

2

1

Mellin Transform

Inverse Mellin Transform

0

1 xdxxs

ssF s

M

ic

ic

s sds

sx

ix

2

1

11

1 xdxxss s

10s.t.integer xxx


Return to TOC

SOLO Primes



nif

nk1

0

11

Möbius Function

The most important property of Möbius function is

10

11

| nif

nifd

nd

The symbol d|n means that the integer d divides the integer n, therefore the sum is on all integers d that divide n. (note that the improper divisor d=1 and d=n have to be included in this formula)

To prove this property, suppose that with all pi being different primes.Then d|n, and μ (d) = (-1)k if d is a product of precisely k different members of the set of s primes pi. This case will occur for different divisors d of n. All divisors d of n containing one or several of the primes pi twice or more have μ (d) = 0, according to the definition of μ (d). Thus

is

i ipn

1

k

s

1,01110|

sifk

sd s

s

k

k

nd

August Ferdinand Möbius1790 - 1868

111|

nd

d

SOLO Primes



nif

nk1

0

11

Möbius Function

The most important property of Möbius function is

10

11

| nif

nifd

nd

Theorem: This relation has as one of its consequence that:

1

1

nsn

n

s

since:

111

1

1

||

1 11

s

ns

nd

s

mdd

m dss

ns n

d

dm

d

d

d

mn

ns

q.e.d.Return to TOC

SOLO Primes

1

1

4

1

3

1

2

1

1

/1

1

4

1

3

1

2

1:

1

0

11

n

n

k

n

n

xn

xxxxxJ



nif

n

xJn

nx

Proof

xdu

xx

umu

xnmm

nm

m

x

n

nxJ

n

n

n ud

u

n um

u

n m

mn

n m

mn

n

n

1 |

/1

1 |

/1

1 1

/1

1 1

/1

1

/1

10

11

| nif

nifd

nd

Conversion from J (x) back to π (x)

q.e.d.

Return to TOC

SOLO Primes

1

1

4

1

3

1

2

1

1

/1

1

4

1

3

1

2

1:

1

0

11

n

n

k

n

n

xn

xxxxxJ



nif

n

xJn

nx

Riemann defined the following formula to approximate the π (x):

The Riemann Prime Number Formula

x

n

n

x

dxxLi

xLixLixLixLixLi

xLin

nxR

0

6/15/13/12/1

1

/1

ln:

6

1

5

1

3

1

2

1

:

SOLO Primes


The Riemann Prime Number Formula (continue – 1)

x

n

n

x

dxxLi

xLin

nxR

0

1

/1

ln:

:

We can see from the Table that R (x) gives a better approximation of the π (x) then Li (x)

SOLO Primes


x

n

n

x

dxxLi

xLin

nxR

0

1

/1

ln:

:

11

ln

1

ln

0

1ln

0

!lnlim

!

lnlnln

!ln

!ln:

n

n

tn

nx

n

n

x

n

neofSeriesTaylor

x tex

dtedx

x

nn

tt

nn

xx

nn

tt

n

dtttd

t

e

x

dxxLi

tt

t

1 !

lnlnln

n

n

nn

xxxLi


SOLO Primes



x

n

n

x

dxxLi

xLin

nxR

0

1

/1

ln:

:

1 !

lnlnln

n

n

nn

xxxLi

1 11

11

1 11

/

1

/1

!

lnln

!

/ln:

n mm

m

nn

n m

m

n

ntex

n

n

mmn

tn

n

nn

n

nt

mm

nt

n

t

n

neLi

n

nxLi

n

nxR

t

0

1limlim

1

11

11

sn

n

n

ns

nss

n

But

1

1

11

1

lim'

lim1

limlim

1lim

lnlim

ln

2

2

12111

1

11

11

1

sos

sos

s

s

ssd

d

n

n

sd

d

nsd

d

n

n

n

nn

n

nn

sssn

ss

nsss

nss

n

SOLO Primes



x

n

n

x

dxxLi

xLin

nxR

0

1

/1

ln:

:

1 !

lnlnln

n

n

nn

xxxLi

1

1

11 1

1 1!1

!1

1

m

mmn

n

nnm n

m

m

mmm

t

n

n

mm

txR

nm

Return to TOC

92

SOLO Primes

Theorem

xn

primepxp

npxk

ln:

For x ≥ 2 we have

otherwise

andpprimesomeforpnifpn

0

1integerln

2/1

22lnln

xOtdtt

t

x

xx

x

Proof

primep

xp

px ln:Define

then

x

xx

x

p

t

ptd

tt

ptd

tt

ptd

tt

t

x

xp

xx

xpxp

x

ptp

xx

tp

x

ln1

ln

ln

ln

ln

ln

ln

ln

ln

lnln/

22

22

22

Von Mangoldt Function

x

tdtt

t

xx

x

xx

x

22ln

1

ln/

Return to TOC

93

SOLO Primes

Hadamard Proof of the Prime Number Theorem (1896)

Hadamard paper on PNT used the Riemann Zeta Function ζ (s) for which he proed some new properties.

His paper published in 1896 consists of two parts:

In the First Part he proved that the Zeta Function has no Zeros on the line Re (s) = σ = 1. His proof is complicated, hence here we give the F. Mertens method to prove this.


(1865 –1963)

94

SOLO Primes

Hadamard Proof of the Prime Number Theorem (continue - 1

Start with the Riemann Zeta Function

11

1lnln1

tispm

psprimep m

ms

SeriesTaylor

primep

s

1

1ln

1

ln1'ln

11

1

tis

pp

p

pp

s

ss

sd

d

primep primep mms

Taylor

p

s

s

s

tispn

sprimep

sn

s

1

1

11

1

132

1ln1

32

xm

x

m

xxxxx

m

m

mmSeriesTaylor

Where the last series counts the prime powers pm, with the weight ln p, therefore

otherwise


k

0


1

1ln

'ln

11

tis

n

n

pp

s

ss

sd

d

ns

primep mms


(1865 –1963)

95

SOLO Primes

Hadamard Proof of the Prime Number Theorem (continue - 2)

otherwise


k

0


1

'

1

tisn

n

s

s

ns Jacques Salomon

Hadamard (1865 –1963)

100

2/12ln

12/112

Re

s

es

ss

es

Hadamard Product Representation of Riemann Zeta Function

Hadamard established the following form of the Mellin Inversion Formula

i

in

sn

s

xnn sd

n

a

s

x

in

xa

2

21

22

1ln

Substitute an = Λ (n)

i

i

si

in

s

s

xn

sds

s

s

x

isd

n

n

s

x

in

xn

2

2 2

2

21

2

'

2

1

2

1ln

96

SOLO Primes

Hadamard Proof of the Prime Number Theorem (continue - 3)


(1865 –1963)

i

i

si

in

s

s

xn

sds

s

s

x

isd

n

n

s

x

in

xn

2

2 2

2

21

2

'

2

1

2

1ln

11

1

tisuduus

s

szd

d

s ReRe

ic

ic

s

sds

x

s

s

ix

'

2

1

primepxpxn k

pnx ln:

1 42

t

10'

xdss

x

s

s

LC

s

LC

R

eRsCj

L

0cos2

2:

R



26 4

Return to TOC

97

SOLO Primes

Newman’s Proof of the Prime Number Theorem (1980)

Proofs have introduced various simplifications to Hadamard and de la Vallée-Poussin through the use of Tauberian theorems but remained difficult to digest, a surprisingly short proof was discovered in 1980 by American mathematician Donald J. Newman. Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis


Prime Number Theorem 1

ln/lim

xx

xx

Newman’s Proof:

x

x

x

x

xx

xxxx

limlimln/

limSince we proved that it is enough to prove that

1lim x

xx

Newman started by proving that

01

2

xdx

xx

First suppose that exists λ > 1 such that θ (x) ≥ λ x for all x sufficiently large (say x ≥ x0)

0

1

0

21

2222

ud

u

uudx

xu

xuxtd

t

txtd

t

tt x

tu

x

tdud

x

x

x

x

Now suppose that exists λ < 1 such that θ (x) ≤ λ x for all x sufficiently large (say x ≥ x0)

This is a contradiction to

01

2

xdx

xx

0

10

2

1

2222

ud

u

uudx

xu

xuxtd

t

txtd

t

tt x

tu

x

tdud

x

x

x

x

This is a contradiction to

01

2

xdx

xx

Therefore the only possibility is:

1lim

x

xx

98

SOLO Primes

Newman’s Proof of the Prime Number Theorem


Newman started by proving that

01

2

xdx

xx

This is done in the following steps:

011

0

111

ln: dteezdx

x

xz

x

x

x

xd

p

pz tzt

ex

dtexdzz

ddvxu

vdxzxduz

primepp

z

t

t

z

z

Newman’s Proof (continue – 1):

Define:

Prove that:

002

12

1 tdeetdee

eexd

x

xx tttt

ttex

tdexd

t

t

1: tt eetf

zz

ztdetdeetdetfzF tztzttz 1

1

1:

00

1

0

???0

1

1

1limlim

00

zz

zzF

zz

Apply Analytical Theorem – A Tauberian Theorem

00

1

1

1limlim

1200

xd

x

xx

zz

zzF

zz

q.e.d.

99

SOLO Primes



011

0

111

ln: dteezdx

x

xz

x

x

x

xd

p

pz tzt

ex

dtexdzz

ddvxu

vdxzxduz

primepp

z

t

t

z

z


Use the Identity:

11

ln

1

ln1ln

tiz

p

p

p

pp

z

zzd

d

zzd

d

primep primepzz

z

111

1

1

zzzz pppp

We found:

11

ln

1

lnln

1

ln

tiz

pp

pz

pp

p

p

p

p

p

z

zzd

d

primepzz

primepzz

primepz

primepz

The sum is:

2/112ln

1

ln2

zzforconvergent

p

p

pp

p

primepz

primepzz

ReRe

100

SOLO Primes




We found:

1

1

ln

1

lnln

1

ln'

tizpp

pz

pp

p

p

p

p

p

z

z

primepzz

primepzz

primepz

primepz

2/112ln

1

ln2

zzforconvergent

p

p

pp

p

primepz

primepzz

ReRe

Change z to z+1:

We found:

We proved also that:

1

1

ln1

1

1

1

1'

foranalyticpp

p

zzz

z

zzz

z

primepzz

0

1

ln

1

11

1

11

11

1'

foranalyticpp

p

zzz

z

zzz

z

primepzz

0

1

1

1lim

0

zz

zz

Stil need to prove

101

SOLO Primes



Analytical Theorem – A Tauberian Theorem


Proof of the Analytic Theorem

t

LC

R

R

R

0

T ts

T dtetfsF0Consider the sequence of functions

Those functions are entire (analytic), and we are trying to show that limT→∞ FT (0) exists and is equal to F (0).

Let chose a closed counterclockwise path of integration γR composed from a semicircle γR

+(z) {z ϵ C| |z|≤ R, Re(z)>-δ}, where we choose δ > 0 small enough (depending on R) so that F (z) is analytic inside γR. (Such a δ exists by compactness and the fact that F (z) is analytic for Re (z) ≥ 0)

Let f (t) be a bounded and locally integrable function for t ≥ 0, and suppose that when Re (z) ≥ 0 extends holomorphically to Re (z) ≥ 0. then exists and equals F (0).

0dttf

0dtetfsF ts

102

SOLO Primes





Proof of the Analytic Theorem (continue – 1)

t

LC

R

R

R

0

Let use the Cauchy Theorem to compute


0dttf

0dtetfsF ts

The additional term z2/R2 was introduced by Newman in order to help the proof.

001

1lim12

12

2

02

2

TzT

Tz

CauchyzT

T FFzR

zezFzFz

z

zd

R

zezFzF

iR

Rz

zd

R

zezFzF

izT

T

2

2

12

1Start with the integral on γR+

zeB

tdetftdetfzFzFTz

T

st

B

tT

stT Re

maxRe

0

2

Re2

*Re

2

22Re

2

2 Re211

R

ze

zR

zzze

zR

zRe

zR

ze TzTzTzzT

R

B

R

ze

z

eBR

z

zd

R

zezFzF

iTz

TzzT

T

R

2Re

Re

2

2 Re2

Re21

2

1

103

SOLO Primes






t

LC

R

R

R

0


0dttf

0dtetfsF ts

RRRz

zd

R

zezF

iz

zd

R

zezF

iz

zd

R

zezFzF

izT

TzTzT

T

2

2

2

2

2

2

12

11

2

11

2

1

Continue with the integral on γR-

Since FT (z) is entire (analytic in all complex plane we can replace γR- with the left

semicircle CL and obtain

R

B

R

ze

z

eBR

z

zd

R

zezF

iz

zd

R

zezF

iTz

Tz

C

zTT

zTT

LR

2Re

Re

2

2

2

2 Re2

Re21

2

11

2

1

104

SOLO Primes






t

LC

R

R

R

0


0dttf

0dtetfsF ts

RRRz

zd

R

zezF

iz

zd

R

zezF

iz

zd

R

zezFzF

izT

TzTzT

T

2

2

2

2

2

2

12

11

2

11

2

1

Continue with the integral on γR-

Finally we observed that the integral converges to zero uniformly on compact sets for Re (z) <0 and T→∞, since the integral is the product of independent of T, and ezT, which goes to zero uniformly on compact subsets of γR.

2

2

1R

ze

z

zF zT

2

2

1R

z

z

zF

012

1lim

2

2

Rz

zd

R

zezF

izT

T

105

SOLO Primes






t

LC

R

R

R

0


0dttf

0dtetfsF ts

tfBR

B

z

zd

R

zezFzF

iz

zd

R

zezFzF

i

z

zd

R

zezFzF

iFF

t

TR

zTT

zTT

zTTT

RR

R

0

2

2

2

2

2

2

max:02

12

11

2

1

12

100

Therefore

0

00lim tdtfFFTT q.e.d.

Return to TOC

106

SOLO

References

Primes

1. Marcus de Sautoy, “The Music of the Primes – Searching to Solve the Greatest Mystery in Mathematics”, Harper-Collins Publishers, 2003

Internet

http://en.wikipedia.org/wiki/

http://www.mathsisfun.com/prime_numbers.html

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/giants.pdf

http://plus.maths.org/content/music-primes

N. Levinson, “A Motivated Account of an Elementary Proof of the Prime Number Theory”, MITK. Chandrasekharan, “Lectures on The Riemann Zeta-Function”, Tata Institute of Fundamental Research, Bombay, 1953

Matt Rosenzweig, “Other Proofs of the Prime Number Theorem”

Jerome Baltzersen, “Hardy’s Theorem and The Prime Number Theorem”, University of Copenhagen, June 2007

G.B. Arfken, H.J. Weber, “Mathematical Methods for Physicists”, Academic Press, Fifth Ed., 2001

107

SOLO

References (continue – 1)

Primes

Internet

B.E. Peterson, “Riemann Zeta Funcyion”, http://people.oregonstate,edu/~peterseb/misc/docs/zeta.pdf

http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html

David Borthwick, “Riemann’s Zeros and the Rhythm of the Primes”, Emory University, November 18, 2009

Ryan Dingman, “The Riemann Hypothesis”, March 12 2010

Laurenzo Menici, “Zeros of the Riemann Zeta-function on the critical lane”, Feb. 4 2012, Universita degli Studi, Roma

P.T. Bateman, H.G. Diamond, “A Hundred Years of Prime Numbers”, http://www.jstor.org

D.J. Newman, “Simple Analytic Proof of the Prime Number Theorem”, The American Mathematical Monthly, Vol. 87, No. 9 (Nov. 1980), pp. 693- 696

M. Baker, D. Clark, “The Prime Number Theorem”, December 24, 2001

http://www.frm.utn.edu.ar/analisisdsys/material/function_gamma.pdf

K.S. Kedlaya, “Analytic Number Theory”, MIT, Spring 2007, “The Prime Number Theorem”

108

SOLO

References (continue – 2)

Primes

Internet

D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf

P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf

http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding2.htm

Robert B. Ash, “Complex Variables”, Chapter 7: The Prime Number Theorem, University of Illinois, http://www.math.uluc.edu/~r-ash/CV/CV7.pdf

Physics 116A, “The Riemann Zeta Function”

M. Rosenzweig, “D.J. Newman’s Method of Proof for the Prime Number Theorem”, M. Rosenzweig, “Other Proofs of the Prime Number Theorem”, http://people.fas.harvard.edu/~rosenzw/

“Notes on the Riemann Zeta Function”, January 25, 2007

109

SOLO

References (continue –3)

Primes

Internet

A. Granville, K. Soundarajan, “The Distribution of Prime Number”

E.C. Titchmarsh, “The Zeta-Function og Riemann”, Cambridge at the University Press, 1980

http://www.dartmouth.edu/~chance/chance_news/recent_news/chance_news_10.10.html

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding2.htm

Hans Riesel, “Prime Numbers and Computer Methods for Factorization”, Chapter 2:“The Primes viewed at Large”,

Prime Numbers and the Riemann Zeta Function « Edwin Chen's Blog

D.R. Heath-Brown, “Prime Number Theory and the Riemann Zeta Function”, http://eprints.maths.ox.ac.uk/182/1/newton.pdf

http://cage.ugent.be/~jvindas/Talks_files/Introduction_Tauberians_Distributional_Approach.pdf

110

Marcus Peter Francis du SautoyProf. Of Mathematics Oxford

University

Return to TOC

April 9, 2023 111

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

http://upload.wikimedia.org/wikipedia/en/a/ae/RAFAEL_logo.png

112

SOLO Primes

Definition of O: (E. Landau Definition)

We say that f (x) = O (g (x)) if exists a constant k > 0 such that |f (x)| < k |g (x)|

Definition of o

We say that f (x) = o (g (x)) when x → a if 0/lim

xgxfax

Asymptotics

Definition axxgxf ,~means

axxgxgxfisthatxgxfax

,,1/lim o

Definitions

113

SOLO Primes

Definition. Let a Function f: Ω → C,

(a)We say that f ϵ C1 (Ω) iff there exists df ϵ C (Ω, M2 (R), a 2x2 matrix-valued function such that

where d f (s) (h) means that the matrix d f (s) acting on the vector h.(b) We say that f is Holomorphic on Ω if

exists for all s ϵ Ω and is continuous in Ω. We denote this by f ϵ H (Ω). A function f ϵ H (C) is called Entire.

0,2 hRhhohsfdsfhsf

Holomorphic, Entire Functions

sw

sfwfsf

sw

lim:'

Note that (b) is equivalent to the existence of a function f’ ϵ C(Ω) so that

where f’(s) h is the product between the complex numbers f’(s) and h. 0,2 hRhhohsfdsfhsf

114

SOLO Primes

Definition.

A Meromorphic Function is a function whose only singularities, except infinity,are poles.

Meromorphic Functions

E.C. Titchmarch, “Theory of Functions” pg. 284b, 110

A Meromorphic Function in a region if is analytic in the region except at a finite number of poles. The expression is used in contrast to Holomorphic, which is some time used instead of Analytic.

Return to TOC

115

SOLO Primes

Mellin Transform

0

1 xdxfxsFxf sMM

We can get the Mellin Transform from the two side Laplace Transform

Robert Hjalmar Mellin ( 1854 – 1933)

xdxfesFxf sx

2LL2

10

11

0

1

sFxdxfxxdxfxxxfx ssMM

ic

ic

s sdsFxi

x M1- fsfM

2

1

Example:

sxdexe xsx

0

1M

xexf

116

SOLO Primes

Mellin Transform (continue – 1)

0

1 xdxfxsFxf sMM

Relation to Two-Sided Laplace Transformation


tdexdex tt ,

Let perform the coordinate transformation

tdeeftdeeftdeefesF tsttstttst

0

1M

After the change of functions teftg :

tdetgsGtdeefsF tstst

2LM

Inversion Formula

xfefsdxsFi

sdesGi

tgxe

tic

ic

sexic

ic

tstt

ML

L

2

12

2

1

2

1

c

t

cxdssFxRC

s 0M

RC

R

117

SOLO Primes

Properties of Mellin Transform (continue – 2)

fkk

k

k

fkk

k

k

fz

fk

kk

fa

f

f

s

SszsFstftd

dt

sksksks

SkszsFkstftd

d

SzszsFCztft

SssFsd

dtft

SsasFaRatf

SsFaataf

SsFtf

HolomorphyofStriptdtftsFtftf

M

M

M

M

M

M

M

MM0t,

1

11:

1

,

ln

0,,

0,11

1

0

1

Original Function Mellin Transform Strip of Convergence

118

SOLO Primes

Properties of Mellin Transform (continue – 3)

21

0

21

1

0

1

0

1

//

1

1

11:

1

11:

1

ff

t

t

k

fkk

k

kk

k

fkkk

k

k

f

s

SSssFsFxxdxtfxf

sFsxdxf

sFsxdxf

kssss

SssFstftd

dt

sksksks

SssFkstfttd

d

SsFtf

HolomorphyofStriptdtftsFtftf

M2M1

M

M

M

M

M

MM0t,

Original Function Mellin Transform Strip of Convergence

Return to TOC

119

SOLO Primes

1Re10

1

zxfordte

tzz

t

tt

z

u

uu

z

due

uz

0

1

Proof:

Gamma Function

Change of variables u=nt

t

tnt

zz

t

tnt

z

tde

tntdn

e

ntz

0

1

0

1

Thus for n=1,2,3,…,N

t

tNt

z

z

t

tt

z

z

t

tt

z

z

tde

t

Nz

tde

tz

tde

tz

0

1

02

1

0

1

1

2

1

1

1

0& xyixz

Summing those equationsfor x > 0

t

t

zNtttzzz

tdteeeN

z0

12

1111

2

1

1

1

_________________________________________________

Proof of Riemann's Zeta Function Relations

120

SOLO Primes

Proof (continue – 1): 0& xyixz

Since converges only for Re (z)= x > 1, then letting N → ∞, we obtain for x > 1

1n

zn

Uniform convergence of

t

t

zNtttNzz

tdteee

z0

12

111lim

2

1

1

1

01

1

1

111

1

2

2

tqeeee t

q

q

t

q

t

q

t

allows to interchange between limit and the integral:

RatioGoldentde

ttd

e

ttd

e

tz

t

tt

zt

tt

zt

tt

z

zz

2

51

1112

1

1

1

ln2

1ln2

0

1

0

1

ln2

02

1ln2

0

1ln2

0

1 11

11

t

ttt

xt

tt

xyixzt

tt

z

tdee

ttde

ttd

e

t

The first integral gives

The integral diverges for 0 < x ≤ 1, and converges only for x > 1

1Re10

1

zxfordte

tzz

t

tt

z


121

SOLO Primes

Proof (continue – 2): 0& xyixz

t

tt

zt

tt

zt

tt

z

zztd

e

ttd

e

ttd

e

tz

ln2

1ln2

0

1

0

1

1112

1

1

1

In the second integral we have

This integral converges only for x > 1, therefore we proved that

ln21 2/ tforee tt

since RatioGoldeneforee ttt

2

5101 2/2/

t

tt

x

termfinite

t

txtu

dtedv

t

tt

xt

tt

xiyxzt

tt

z

tde

txettd

e

ttd

e

ttd

e

t x

t

ln22/

2

ln2

2/1

ln22/

1

ln2

1

ln2

1

12211

1

2/

finite

t

ttxx

xxt

tt

x

tdet

xxxxtermsfinitetde

t

ln22/1

ln22/

1 1212

1Re12

1

1

1

0

1

zxfortde

tzzz

t

tt

z

zz

1Re10

1

zxfordte

tzz

t

tt

z

After [x] (the integer defined such that x-[x] < 1) such integration the power of t in the integrand becomes x-[x]-1 < 0. and we have:

q.e.d.


Return to TOC

122

SOLO Primes

Proof

The integral can be rewritten as

00

1sin2

1 0

0

1

itoreturnsandzeroencirclesiatstartspaththe

de

izz

zi

i

z

i

iy

x

i

i

2

IntegralIII

i

i

z

originaroundCircleIntegralII

i

i

z

IntegralI

i

i

z

i

i

z

de

de

de

de

1lim

1lim

1lim

11

0

1

0

1

0

0

0

1


123

SOLO Primes

Proof (continue – 1)

The first integral can be written as

i

iy

x

i

i

2

t

tt

zzi

et

tt

zzi

t

tit

ziiti

i

z

tde

tetd

e

tetd

e

eitd

e

i

0

110 11

1

0

0 1

0 111lim

1lim

The second integral can be written as

0

1

2lim2

1

2lim

21

2lim

1lim

2

020

2

02

1

0

2

02

1

0

21

0

de

deie

e

deie

ed

e

ii

i

i

e

x

i

e

iyxi

i

e

iyxie

originaroundCircle

i

i

z

00

1sin2

1 0

0

1


de

izz

zi

i

z


124

SOLO Primes


The third integral can be written as

i

iy

x

i

i

2

t

tt

zzi

et

tt

zzi

t

tit

ziiti

i

z

tde

tetd

e

tetd

e

eitd

e

i

0

11

0

11

1

0

1

0 111lim

1lim

Therefore

t

tt

zt

tt

zzizit

tt

zzizi

i

i

z

tde

tzitd

e

t

i

eeitd

e

teed

e 0

1

0

1

0

10

0

1

1sin2

122

11

But we found that 1Re10

1

zxfordte

tzz

t

tt

z

00

1sin2

1 0

0

1


de

izz

zi

i

z

Therefore

0

0

1

1sin2

1 i

i

z

de

izz

z

The right hand is analytic for any z ≠ 1. Since it equals Zeta Function in the half plane x > 1, it is the Analytic Continuation of Zeta to the complax plane for any z ≠ 1.

0

0

1

1sin2

1 i

i

z

de

izz

z

q.e.d.


Return to TOC

125

SOLO Primes

Proof

0

0

1

12

sin221

i

i

zz

zz

ide

R

RC

C

plane

Re

Im

Let add a circular path of radius R → ∞. On this path

0

1lim

1

2

0

1

d

e

eRd

ei

i

i

R

eR

zi

R

eR

deRdC

z

Therefore we have

d

ed

ed

ed

e

z

C

zi

i

zi

i

z

R1111

110

0

10

0

1

Since the integral is over a closed path in the complex λ plane, we can use the Residue Theorem to calculate it. The residues are given by

,2,121 nnie

1

1

1

110

0

1

222211 n

z

n

zzi

i

z

niiniide

de


126

SOLO Primes

Proof (continue)

0

0

1

12

sin221

i

i

zz

zz

ide

R

RC

C

plane

Re

Im

11

11

1

1

1

10

0

1 122222

1 nz

zzz

n

z

n

zi

i

z

niiiniiniid

e

2sin22/2/lnln11 z

eeieeiiiiiii izizizizzzzz

znn

z

1

1

11

zzid

ez

i

i

z

12

sin221

0

0

1

q.e.d.


Return to TOC

127

SOLO Primes

i

iy

x

i

i

2

00

12

1 0

0

1


dei

zz

i

i

z

We also found

zzid

ez

i

i

z

12

sin221

0

0

1

Has zeros for

,...4,2,002

sin

zforz

,7,5,301 zforz

z 1 Has no zeros, but has simple poles for z = 1,2,3,4,….

If we return to ζ (z) equation we can see that the zeros of are cancelled by the poles of Γ (1-z). Only the simple pole at z = 1 remain and is the single pole of ζ (s).

0

0

1

1

i

i

z

de

Let find the Residue of this pole:

0

0

1

111 12

1lim11lim1lim

i

i

z

zzzd

eizzzz

1cos

limsin

1lim11lim

1

'

1

1

sin1

1

zzz

zzz

z

HopitalL

z

zzz

z

0

0

1

1 12

1lim

i

i

z

zd

ei


Return to TOC

128

SOLO Primes

Proof

zzzzz z

1

2sin22sin2

zzid

ez

i

i

z

12

sin221

0

0

1

We found

0

0

1

1sin2

1 i

i

z

de

izz

z

Combining those two relations, we get

zzzzz z

1

2sin22sin2

q.e.d.


Return to TOC

129

SOLO Primes

Proof

zzzz zz 112/sin2 1

Start from

use

zzzzz z

1

2sin22sin2

zzz

sin1 zzz

1sin

or

zzz

zz

1

2sin2

1

zzzz zz

1

2sin12 1

q.e.d.

Proof of Riemann's Zeta Function Relations Return to TOC

Return to Riemann Zeta Function

130

SOLO Primes

Proof

Start from

use

zzzzz z

1

2sin22sin2

zzz

sin1

zzz

1

sin

21

22

sinzz

z z

z

2

zz

zzz

zz

1

2sin

2/12/

12

1

1

or

zz

zzz zzz

12/1

1122/ 2/12/12/

z

z

z

z zzzz

1

2/12/ 12/12/


131

SOLO Primes

Proof (continue)

z

z

z

z zzzz

1

2/12/ 12/12/

or

zz

zzz zzz

12/1

1122/ 2/12/12/

2

1

22 12/1 zz

z z

2/12

121 2/1 z

zz z

z

z

1

2

1

2/1

1122/1 z

zzz

therefore

q.e.d. zz

zz zz

1

2

12/ 2/12/

Use LegendreDuplication Formula: 0Re2

22

112

zzzz

z

2/z

z


132

Jacob Bernoulli1654-1705

The Bernoulli numbers are among the most interesting and important number sequences in mathematics. They first appeared in the posthumous work "Ars Conjectandi" (1713) by Jakob Bernoulli (1654-1705) in connection with sums of powers of consecutive integers. Bernoulli numbers are particularly important in number theory, especially in connection with Fermat's last theorem (see, e.g., Ribenboim (1979)). They also appear in the calculus of finite differences (Nörlund (1924)), in combinatorics (Comtet (1970, 1974)), and in other fields.

Bernoulli Numbers The Bernoulli numbers Bn play an important role in several topics of mathematics. These numbers can be defined by the power series

SOLO

0 !1 n

n

nz n

zB

e

z

133

SOLO Primes

Bernoulli Numbers

0 !1 n

n

n

seriesTaylor

z n

zB

e

z

Let compute the Bernoulli number using

1Residue2

2

!

12

!

11

z

n

eCnzn e

zi

i

n

z

zd

e

z

i

nB

z

R

RC

C

planez

zRe

zIm

The zeros of e z = 1 are at z = ± 2 π i k

1 1

1 12

1'

22

1'

2

1

2

1

2

1!

1lim

1

2lim

1lim

1

2lim2

2

!

1Residue2

2

!

k knn

k knkiz

Hopitall

zkiznkiz

Hopitall

zkiz

z

n

en

kikin

ze

kiz

ze

kizi

i

n

e

zi

i

nB

z

01

x

xn

n

n e

x

xd

dB

134

SOLO Primes

Bernoulli Numbers

0 !1 n

n

nz n

zB

e

z

Let compute the Bernoulli number using

1Residue2

2

!

12

!

11

z

n

eCnzn e

zi

i

n

z

zd

e

z

i

nB

z

R

RC

C

planez

zRe

zIm

The zeros of e z -1 = 1 are at z = ± 2 π i k

1 11 2

1

2

1!

1Residue2

2

!

k knnz

n

en

kikin

e

zi

i

nB

z

oddn

evennk

oddn

evennki

kikik

nn

k

n

kn

kn

0

12

0

2111

2/

111

oddn

evennnn

k

nB n

n

knn

n

n

0

2

!12

1

2

!12

2/

1

2/

,2,1,0

120

222

!212 2

m

mn

mnmm

B m

m

n

Return to Riemann Zeta Function Return to Riemann Zeta Function

135

SOLO Primes Bernoulli Numbers

0 !1 k

k

nz k

zB

e

zLet multiply with

1 !

1m

mz

m

ze

2

1

00

2

1

00

variablesofChange

10

2

01

!

1

!

!

1

!

!!

1

!!

1

11

n

nn

k

k

n

n

k

knkk

knm

k

k

nm

m

k

k

nm

mz

z

zknk

BzB

zknk

BzB

k

zBBz

mz

k

zBz

me

zez m

k

knm

0, knm

0

1,1 nkm

Therefore, equalizing the coefficients of z, from the two side we obtain

10 B 20

!!

11

0

nBknk

n

kk

Algorithm for Bernoulli Numbers Computation

136

SOLO Primes Bernoulli Numbers

0 !1 k

k

nz k

zB

e

zLet multiply with

1 !

1m

mz

m

ze

We obtained 10 B

20!!

!

!!

1!

1

0

1

0

1

0

nBk

nB

knk

nB

knkn

n

kk

n

kk

n

kk

This demonstrates that the Bernoulli numbers are rational, and we have an algorithm to calculate the nth Bernoulli number.

2

1

2

10

!1!1

!2

!2!0

!22 011

1

0

1

BBBBn

6

1

2

31

3

10

!1!2

!3

!2!1

!3

!3!0

!33 22

3

2/1

1

3

1

0

1

BBBBn

Algorithm for Bernoulli Numbers Computation (continue)

Return to TOC

137

SOLO Primes

Proof

i

iy

x

i

i

2

00

1sin2

1 0

0

1


de

izz

zi

i

z

We found

and zzz

sin1 zzz

sin1

0

0

12

1 0

0

1

itoreturnsand

zeroencirclesiatstartspaththe

dei

zz

i

i

z

0 1

21 12

!1

12

!1 i

i

n

Cnzn d

ei

n

z

zd

e

z

i

nB

i

iy

x

i

i

2

therefore 1!1

11

n

n Bn

zz

0

0

12

1 0

0

1

itoreturnsand

zeroencirclesiatstartspaththe

dei

zz

i

i

z

zz

nz 11 1

n

Bn nn

,2,11

1 1

nn

Bn nn Bn are the Bernoulli numbers

q.e.d.

We found

Zeta-Function Values and the Bernoulli Numbers


138

SOLO Primes

Zeta Function Values and the Bernoulli Numbers

,2,1,0

!22

212 2

2

mBm

m m

mm

zzzz zz

1

2

12/ 2/12/ Let use

with z = 2 m mm

mm mm 212

212 2/21

m

mm

m

mmm

m

m

Bmm

mB

mm

m

mm

mm

2

2/112

2

22/12

2/1

2/1!2

!121

!22

21

2/1

!1

22/1

!121

We found

,3,2,1

2/1!2

!12121 2

2/112

mBmm

mm m

mm

139

SOLO Primes

Zeta Function Values and the Bernoulli Numbers

We found

2/112

2/11

2/1

!12

!121

224212531

121221

12531

212/1

m

m

mm

m

mm

mm

mmmmm

,3,2,1

2/1!2

!12121 2

2/112

mBmm

mm m

mm Therefore

Finally

,2,11

1 1

nn

Bn nn

,3,2,12

121 2 mB

mm m

We also found The two expressionsAgree.

Return to TOC


140

SOLO Primes

zzz z 2/: 2/ zz 1We found that

The function η (z) has simple poles at z = 0 of Γ (z/2) and one at z = 1 of ζ (z).

A new function, where those poles are removed ( entire = analytic in all complex plane) and retains the symmetry relation is

zzzzz z 2/12

1: 2/ zz 1

We saw that ζ (z) has trivial zeros at z = -2 n , n=1,2,3,…. Also Γ (z/2) has simple poles at z/2= -1,-2,-3….. Thus for z/2=-1,-2,-3,… the simple poles of Γ (z/2) cancel the trivial zeros of ζ (z).

Therefore the zeros of ξ (z) = 0 are equal to the non-trivial zeros of ζ (z) = 0.

2

1101lim

22lim2/1

2

1limlim0

12/

0

11

21

0

2/

00

zzz

z

z

z

z

zzzz

zzzzzzz

2

12/1

2

12/

2

1lim1lim2/1

2

1limlim1

2/1

2/12/

1

1

1

2/

11

zzzzzzzzz z

zz

z

zz


141

SOLO Primes

zzzz z 2/11: 2/ zz 1The zeros of ξ (z) = 0 are equal to the non-trivial zeros of ζ (z) = 0.

For Re {z}=σ > 1, we have

1Re01

1

tizfor

pz

primepz

102/112/

Therefore if ρ is a zero of ζ (z) , so is 1 - ρ i.e. ζ (1-ρ) = 0, and

1Re111Re

The zeros of ξ (z) that are equal to the non-trivial zeros of ζ (ρ) = 0 can exist only on the Critical Strip: 0 < Re (ρ) < 1.


01012/2/310

2/1

Suppose ρ is a zero of ζ (z) , i.e. ζ (ρ) = 0 and, therefore, Re {ρ} < 1. Re {ρ} < 1

Since the product contains no zero factors we see that ζ (z) ≠ 0 for σ = Re {z} >1.σ = Re {z} >1

Re0

142

SOLO Primes

zzzz z 2/11: 2/ zz 1The zeros of ξ (z) = 0 are equal to the non-trivial zeros of ζ (z) = 0.

The zeros of ξ (z) that are equal to the non-trivial zeros of ζ (ρ) = 0 can exist only on the Critical Strip: 0 < Re (ρ) < 1.


SOLO Primes

zzzz z 2/11: 2/ zz 1Zeros of Zeta-Function: ζ (z) = 0

• ξ (z) is an entire function (analytic in all the complex plane except at ∞)

ξ (z) is regular for Re {z}=σ > 0

ξ (1-z) is regular for Re {1-z}=1-σ > 0, i.e. σ < 1. Since ξ (z)= ξ (1-z) it is regular for both σ > 0 and for σ < 1, meaning for all σ, i.e. ξ (z) is entire.

SOLO Primes


• ξ (z) has no zero at the Re{z} = σ = 1

tizpn

zprimep

zn

sRe

1

11

1

Start with

0cos12coscos2121cos2cos432coscos43 222 Suppose that ζ (1+i t) = 0, and consider (Mertens 1898)

tititD 2:, 43 At z=1 the pole of ζ (z) at z = 1 will cancel some of the alleged ζ (1+i t), and 1 + 2 i t may be a zero of ζ (z) but is certainly not a pole. Therefore, if we prove that D (z) does not have a zero at z = 1, than we prove that ζ (z) has no zero on the line Re{z} = σ = 1 line.

Proof:

11

1lnln1

tizpm

pzprimep m

mz

SeriesTaylor

primep

z

tizpm

ppzprimep m

mz

SeriesTaylor

primep

z

primep

z

1

1Re1lnRe1lnln

where we used 1/1ln1

wnwwn

n

SOLO Primes



tizpn

zprimep

zn

sRe

1

11

1

tititD 2:, 43


tizpm

ppzprimep m

mz

SeriesTaylor

primep

z

primep

z

1

1Re1lnRe1lnln

Hence

primep m

mtiimtm

primep m

imtm

primep m

imtm

primep m

m

pppm

pm

pm

pm

tititD

1

2

1

2

11

43Re1

1Re

1Re4

1Re3

2lnln4ln3,ln

pmtipmteppmtipmtep pmtiimtpimtimt ln2sinln2cos&lnsinlncos ln2ln

pmtpm

tD mprimep m

mmm

m

ln:02coscos431

,ln1

cos12 2

SOLO Primes



tizpn

zprimep

zn

sRe

1

11

1

tititD 2:, 43 Proof (continue – 2):

pmtpm

tD mprimep m

mmm

m

ln:02coscos431

,ln1

cos12 2

Therefore: 12,43 tititD

1

1lim2lim

1lim1lim

1

,lim

0101

4

01

3

0101

ti

titD

Let find:

Since ζ (z) has a simple pole at z=1

11lim01

If ζ (1+it) has a zero for a t then

1lim

01

it

We see that the left side of the inequality above is finite when σ→1, but the right side is infinite, a contradiction. We conclude that ζ (1+it) ≠ 0. Since t is arbitrary, ζ has no zeros on Re {z} = σ = 1. q.e.d.

Return to TOC

SOLO Primes

The Behavior of ζ (x) as x → 1

1,1ln1

2ln

12ln

2

1

1

1

2

xasxnnx

xn

n

O

xxxx

nx

xxxxx

xxxxxxxxxn

x

n

11

1

2123

1

2

11

2

2

6

1

4

1

2

12

4

1

3

1

2

11

4

1

3

1

2

11

1:

Proof:We have: 1

1

1

xn

xn

x

Define the Dirichlet Eta Function:

Therefore:

1

11

1

21

1

21

1

nx

n

xx nxx

2ln2

11

2ln

1

2ln21

1

1

2ln

1

2ln21

2ln11

1

1

1

21

1

21

1

222ln

1

1

2ln

eTaylorx

x e


SOLO Primes

The Behavior of ζ (x) as x → 1

1,1ln1

2ln

12ln

2

1

1

1

2

xasxnnx

xn

n

O


1

1

1

21

1

nx

n

x nx

2ln2

11

2ln

1

21

1 1

1

x

x

nnnnennnnn n

x

xln1

1

ln1

11111ln1

1

2

1111

1

1

ln1

2lnln1111

O

n

n

n

n

n

nx

nx

n

nn

nnnn

Therefore using we obtain:

1

1 /12lnn

n n

1ln1

2ln

12ln

2

11

ln1

2ln2ln2

11

2ln

11

2

2

2

xnn

nn

n

n

n

n

O

O

q.e.d.


149

SOLO Primes

Proof

11

11

1

1

tispn

sprimep

sn

s

11

1

1

11

1

1

111

n

n

n

sn

n

s

n

n

n

s

n

ss

n

s dxxdxndxxndsxns

s

1

1

ss extends to an analytic function in the right half plane (σ > 0)

For σ > 1,

Fix n ≥ 1. Observe that for x ϵ [n,n+1],

11

11

nduuduusxnn

n

x

n

sss

Since converges for all σ > 0, we see that

1

1

nn

11

1

1

1

n

n

n

ss xdxns

s

converges uniformly in any closed half plane { σ ≥ ε}, for ε > 0. Thus, the series defines an analytic function in the right half plane { σ ≥ ε}. q.e.d.


Theorem

150

SOLO Primes

Proof

11

11

1

1

tispn

sprimep

sn

s

01

1:ˆ

foranalytic

sss

1

1'

ss

s

extends to an analytic function in the right half plane (σ ≥ 1)

Define:

q.e.d.

ss

sss

ss

sssss

ss

sss

ss

s

1

ˆ'ˆ1

1

1ˆ11'ˆ

1

1'

1

1' 12

01

1':'ˆ 2

foranalytic

sss

01

11lim1lim:ˆResidue 1s

1

1s1s

sssss

01

1ˆ

1

ˆ'ˆ11lim

1

1'Residue

11s

ss

ssss

ss

ss

Return toIkehara-Wiener Theorem


Theorem

Return to TOC

151

SOLO Primes

Theorem

Proof

11

11

1

1

tispn

sprimep

sn

s

01

1:ˆ

foranalytic

sss

1

1'

sss

s


Define:

q.e.d.

sss

ssss

sss

ssssss

sss

ssss

sss

s

1

1ˆ'ˆ1

1

1ˆ11'ˆ

1

1'

1

1' 12

01

1':'ˆ 2

foranalytic

sss

01

11lim1lim:ˆResidue 1s

1

1s1s

sssss

01

11ˆ

1

1ˆ'ˆ11lim

1

1'Residue

11s

sss

sssss

sss

ss

Return to Newman’a Proof


SOLO Primes

Zeta Function ζ (s) and its Derivative ζ‘ (s)

11

lim

1:

1

1

forconvergestisn

ss

ns

nsN

N

N

nsN

1ln

'lim'

ln:'

1

1

tisn

nss

n

ns

sd

ds

nsN

N

N

nsNN

sp

p

n

ns

primeps

ns

:lnln

'1

1

1

ln1lim

ln1:

ns

kkk

NN

k

N

ns

kk

Nk

kk

N

n

nss

n

ns

sd

ds


Return to TOC

153

SOLO Primes

11

1

tizduuuz

z

zzd

d

z ReRe

Theorem

Define:

Proof

11

ln1

ln1ln

11

1

tiz

pp

p

pp

z

zzd

d

zzd

d

primep primep mmz

Taylor

p

z

z

z

otherwise


k

0


The infinite sums are convergent for σ > 1 so we can write

primep k

zpk

primep m

zm

primep mmz

pkppp

pm

lnln1

ln11

11ln111

tizkkkkkpk

z

zzd

d

k

z

k

z

primep k

z

xn

nx :


154

SOLO Primes

11

1

tizuduuz

z

zzd

d

z ReRe

Theorem


11ln111

tizkkkkkpk

z

zzd

d

k

z

k

z

primep k

z

Let compute

zzM

k

zM

k

zM

k

zz

M

k

zzM

k

zzM

k

zM

k

zM

k

z

kkkMMkkkkMM

kkkkMMkkkkkkk

11

111111

1

1

1

1

1

1

20

1

1111

We have 0lnlimlimlim

Mn

primepMp

z

MMn

z

M

z

M k

pMnMMM

1

1

11

1

11

1

1

1

11

1

1

1M

zM

k

k

k

zku

kuk

M

k

k

k

zzzM

k

uduuzuduuzudukzkkk


155

SOLO Primes

11

1

tizuduuz

z

zzd

d

z ReRe

Theorem


1lim1limlim1

11

1

11

10

uduuzuduuzkkkMM

z

zzd

d

zM

z

M

zzM

kM

z

M

q.e.d.


Return to TOC

156

SOLO Primes

1'

1

1

tizuduuz

z

z

z

zzd

d

z ReRe

We found

0

1 udufuvFvf vMM

ic

ic

s sdsFvi

vvf M1- fM

2

1

ic

ic

z

zdz

x

z

z

ix

'

2

1

The Mellin and the Inverse Mellin Transform of a function f (z) are defined by

From the relation above we can see that is the Mellin Transformation of ψ (x).Therefore we can use the Inverse Mellin Transform to obtain:

zz

z

'



Return to TOC

157

SOLO Primes

Proof

tispn

sprimep

sn

s

1

1

11

1

2

11ln1

11lnln

nss

primep nnn

ps

1,

1ln

2

tisxdxx

xss

s

primepxp

xprimesofnumberx 1:

The last equality is true since

otherwise

primenifnn

0

11

21

00

2 1

11ln

11ln1

11ln1

ns

ns

ns n

nn

nn

n

using

we obtain

2 1

11ln

11lnln

nss nn

ns


158

SOLO Primes

Proof (continue)

1,

1ln

2

tisxdxx

xss

s

primepxp

xprimesofnumberx 1:

2 1

11ln

11lnln

nss nn

ns

Let compute

sss

s

s

s

s

ss

n

n

sn

n

n

ns

sn

ns

nnn

n

n

n

n

n

n

n

xxsxdx

xs

x

sxd

xx

s

1

11ln

11ln

11

11ln

1

11ln

1ln

1lnln11

111 11

therefore 11

ln2

1

n

n

ns

xdxx

sns

hence 1

1ln

2

xdxx

xss

s

q.e.d.


Return to TOC

159

SOLO Primes

The Riemann Zeta Function can be factored over its nontrivial zeros ρ as the Hadamard Product:

/

2/12ln

1

21

112: ze

z

zz

ez

Hadamard (1893) used the Weierstrass product theorem to derive this result. The plot above shows the convergence of the formula along the real axis using the first 100 (red), 500 (yellow), 1000 (green), and 2000 (blue) Riemann zeta function zeros.

γ is the Euler-Mascheroni constant

γ=0.57721566490153286060651

Hadamard Product of ζ (s)


(1865 –1963)

1

22

1

2/

010

21

22/

1

211

1:

n

n

zz

n

n

s

sofzerostrivial

xsba

en

ze

z

z

en

se

s

s

es

zzz

2

1

2

12

2

112


Return to TOC

160

SOLO Primes

Perron’s Formula

11

10

2

1

2:Re aif

aifds

s

a

i ss

s

Oskar Perron

( 1880 – 1975)

1 32

t

10 adss

a

LC

s

10 adss

a

RC

s

LC

RC

R

eRsCj

L

0cos2

2:

R

eRsCj

R

0cos

2:

R

Proof

Define the two semi-circular paths CL (left side), CR (right side) with s=2 as the common origin., and R → ∞.

RLRL

RLRL

C

R

C

R

C

iiRR

C

s

dadRR

a

deRiiRR

ads

s

a

,,

,,

coscos

sincos

sincos

LR

LR

RLRL

CC

CC

C

R

RC

s

R aora

aora

dadss

a

)0cos&1()0cos&1(

)0cos&1()0cos&1(0

limlim,,

cos

161

SOLO Primes

Perron’s Formula

11

10

2

1

2:Re aif

aifds

s

a

i ss

s

1 32

t

10 adss

a

LC

s

10 adss

a

RC

s

LC

RC

R

eRsCj

L

0cos2

2:

R

eRsCj

R

0cos

2:

R

Proof (continue)

We can see tat

10Residue

11lim

1Residue

1Residue

2Re

0

2Re

2Re

as

a

as

as

as

a

as

a

s

Cs

s

s

s

Cs

s

Cs

RR

L

q.e.d.

1Residue

1Residue

1

1

1

1

2

1

2Re

2Re

2Re

2Re

0

2

22:Re as

a

as

a

adss

a

adss

a

adss

a

adss

a

dss

ads

s

a

i s

Cs

s

Cs

Cs

s

Cs

s

C

s

C

s

i

i

s

ss

s

R

L

R

L

R

L

Return to TOC

SOLO Primes

Alfred Tauber ( 1866 – 1942)

Auxiliary Tauberian Theorem

Abel Theorem on Power Series (1826).If the Series is convergent to the number β then

0n nc

01

limn

nn

rrc

Niels Henrik Abel ( 1802 – 1829)

One other statement of Abel Theorem: A possible divergent series is said to be Abel summable to β if the power series has radius of convergence at least 1 and .

0n ncn

n nrc

0

01lim

n

nn

rrc

The genesis of Tauberian Theory is do to Alfred Tauber, that proved in 1897 the following Theorem that gives a sufficient condition for the converse of Abel’s Theorem

Tauber Theorem on Power Series (1897).If and

0n

nnrc

then the Series converges to the number β

0n nc

n

ncn ,

1o

Definition of oWe say that f (x) = o (g (x)) when x → a if 0/lim

xgxf

ax

http://en.wikipedia.org/wiki/File:Niels_Henrik_Abel.jpg

SOLO Primes

Auxiliary Tauberian Theorem (continue – 1)


Godfrey Harold "G. H." Hardy

] ( 1877 – 1947)

The phrase “Tauberian Theorem” was coined by G.H. Hardy,who along with J.E. Littlewood made a number of contributions in this area.

Generally “Tauberian Theorems” are those in which some type of “ordinary” convergence (e.g. convergence ofeach ), is deduced from some “weaker” type of convergence (e.g. convergence of for each z with Re z>0) provided additional conditions are satisfied (e.g. has an analytic extension to a neighborhood of the each point on the imaginary axis.

0dtetF tyi

Ry

0dtetF tz

0dtetFzG tz

Return to Analytic Theorem

http://en.wikipedia.org/wiki/File:[email protected]

SOLO Primes


In 1910 Littlewood gave a weaker condition for convergence then Tauber where he replaced o (1/n) with O (1/n).

then the Series converges to the number β

0n nc

Littlewood Theorem on Power Series (1910).If and

0n

nnrc

n

ncn ,

1O


Godfrey Harold "G. H." Hardy

] ( 1877 – 1947)

Two years after Hardy and Littlewood conjectured and later proved the following:

Hardy and Littlewood TheoremLet be convergent for |r| < 1. Suppose that for some number α ≥ 0

0n

nnrc

Crcrn

nn

r

0

1lim

If 1 nallfornMcn n

then

1

','0

CCwherenCc

nn

kk


The phrase “Tauberian Theorem” was coined by G.H. Hardy.

http://en.wikipedia.org/wiki/File:[email protected]

SOLO Primes


In 1931 Ikehara, a student of Norbert Wiener, showed the following Tauberian Theorem for Dirichlet Series. His aim was to give a proof of the Prime Number Theorem (PNT) in the form

1lim x

xx

Ikehara-Wiener Theorem (1931)

Let F be given by the Dirichlet Series

1,1

sforconvergentn

csF

nsn Re

where the coefficients satisfy the Tauberian condition cn ≥ 0. If there exists a constant β such that

1

z

zF

admits a continuous extension to the line Re z = 1, then

N

nn

Nc

N 1

1lim

Shikao Ikehara (1904 –1984)

Norbert Wiener ( 1894 – 1964 )

http://en.wikipedia.org/wiki/File:Norbert_wiener.jpg

SOLO Primes

Auxiliary Tauberian Theorem (continue – 4)Ikehara-Wiener Theorem (1931)

Let F be given by the Dirichlet Series 1,1

sforconvergentn

csF

nsn Re

where the coefficients satisfy the Tauberian condition cn ≥ 0. If there exists a constant β such that

1

zzF

admits a continuous extension to the line Re z = 1, then

N

nn

Nc

N 1

1lim

Proof of PNT

1

1ln

'ln:

11

tis

n

n

pp

s

ss

sd

dsF

ns

primep mmsChoose

1

1'

ss

s


Therefore ncn :We proved

According to Ikehara-Wiener Theorem with β = -1

1lim

1lim

1lim

1

x

xn

xc

N xxn

x

N

nn

N

1/lim

xxx

q.e.d.

SOLO Primes


Tauberian LemmaLet f (x) be a function on [2,∞) and suppose x f (x) monotone non-decreasing on [2,∞). Let m and n be real numbers, n ≠-1. If

then

xx

xtdtf m

nx

ln~

1

2

xx

xnxf m

n

ln

1~

Proof:

Let x ≥ 2, ε > 0 and x (1-ε) ≥ 2

111

1lnx

x

x

x

x

x

xfxt

tdxfx

t

tdtfttdtf

and

x

x

x

x

x

x

xfxt

tdxfx

t

tdtfttdtf

111

1ln

SOLO Primes



Let ε > 0. By hypothesis there exists Aε ≥ 2 such that x ≥ Aε

m

n

m

nnxxx

x x

x

x

xtdtftdtftdtfxfx

ln

1

1ln

111ln

21211

2

1

2

1

xx

xtdtf

x

xm

nx

m

n

ln

1

ln

1 21

2

21

thus if 0 < ε < 1 and then

Ax ,

1

2max

mnn

m

nxxx

x x

x

x

xtdtftdtftdtfxfx

1ln

11

ln

11ln

211211

221


then

xx

xnxf m

n

ln

1~

xx

xtdtf m

nx

ln~

1

2

SOLO Primes




then

xx

xnxf m

n

ln

1~

xx

xtdtf m

nx

ln~

1

2

If follows that if 0 < ε < 1 then

1ln

111lnsuplim

221n

n

m

x x

xfx

1ln

111lninflim

212 n

n

m

x x

xfx

11/1lim

221111lim

1ln

111lim

0

12

0'221

0

n

n nnHopitalLn

11/1lim

211112lim

1ln

111lim

0

12

0'212

0

n

n nnHopitalLn

SOLO Primes




then

xx

xnxf m

n

ln

1~

xx

xtdtf m

nx

ln~

1

2

If follows that as x → ∞ then

1

lnlim

n

x

xfxn

m

x

therefore

q.e.d.

m

n

mnx x

xnxf

xxn

xf

ln

1~1

ln/1lim

SOLO Primes


Chebyshev Function Method

Since

0

1'xdxx

ss

s s

Using Mellin’s Inverse Transform we have

ic

ic

s sdxss

s

ix

'

2

1

By the contour integral we obtain the Psi von Mangoldt formula of 1895

1 42

t

10'

xdss

x

s

s

LC

s

LC

R

eRsCj

L

0cos2

2:

R



26 4

2/12

0Re0

1

1ln0

0'ln:

xx

xpx

mprimep

xpm

We then use this formula to deduce 2

~2

1

xtdt

x

This part uses the fact that are no roots on Re s = 1. Then the Tauberian Lemma, yields xx ~There are some variants of this method:1.Hadamard estimated

2.La Valée Poussin estimated

xtd

t

t1 2

xtd

t

t1

SOLO Primes


La Valée Poussin Function Method

Define

0

1'xdxx

ss

s s

Using Mellin’s Inverse Transform we have

ic

ic

s sdxss

s

ix

21

'

2

1

1 42

t

10'

xdss

x

s

s

LC

s

LC

R

eRsCj

L

0cos2

2:

R



26 4If we estimate the contour integral we obtain ψ1 (x) ~ x. Then the Tauberian Lemma, yields.

10:1

xtd

t

tx

1~

x

x

ic

ic

s sdxss

s

ix

'

2

1

ic

ic

sic

ic

xs

xic

ic

sx

sdss

x

ss

s

itdtsd

ss

s

it

tdsdt

ss

s

idt

t

t

'

2

1'

2

1'

2

1 1

Given Mellin Transform of Chebyshev ψ (x) and its Inverse:

Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin

(1866 1962)

SOLO Primes


Riemann’s Function MethodRiemann didn’t prove the Prime Number Theorem but he did have the ingredients described here

0 1

lnxd

xx

x

s

ss

Since

1

1lnxdxxJ

s

s s

1

1

4

1

3

1

2

1 1

4

1

3

1

2

1:

n

nxn

xxxxxJ

Then using Mellin’s Inverse Transform

ic

ic

s sdxs

s

ixJ

ln

2

1

This integral is only conditionally convergent.We estimate and deduce

Then

xxxJ ln/~

xxxxxxxJx ln/ln/ln/2/1 oO

and

Return to TOC

174

SOLOInfinite Series

Given a series:

Theorems of Convergence of Sequences and Series

n

iin uS

1Convergence Definition:

The series Sn converges to S as n →∞ if for all ε > 0 there exists an positive integer N such that

If no such N exists then we say that the series diverges.

NnallforSuSSn

iin

1

Convergence Theorem:The series Sn converges as n →∞ if if and only if there exists an positive integer M such that

If no such M exists then we say that the series diverges.

11

NallforMuSN

iiN

If S is unknown we can use the Cauchy Criterion for convergence: for all ε > 0 there exists an positive integer N such that

NmnallforuuSSm

jj

n

iimn

,11

Augustin Louis Cauchy ) 1789-1857(

A necessary (but not sufficient) condition for convergence is that lim i→∞ ui = 0

175

SOLO Complex Variables

Let {un} :=u1 (z), u2 (z),…,un (z),…, be a sequence of single-valued functions of z insome region of z plane.

We call U (z) the limit of {un} ,if given any positive number ε we can find a number N (ε,z) such that and we write this: zNnzUzu n ,

zUzuorzUzun

nnn

lim

x

y C

R

If a sequence converges for all values z in a region R, we call Rthe region of convergence of the sequence. A sequence that is notconvergent at some point z is called divergent at z.

Infinite Series

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176

SOLO Complex VariablesInfinite Series

Series of Functions From the sequence of functions {un} let form a new sequence {Sn} defined by:

n

i

inn zuzuzuzuzS

zuzuzS

zuzS

1

21

212

11

If , the series is called convergent and S (z) is its sum. zSzS nn

lim

A necessary (but not sufficient) condition for convergence is that lim n→∞ un(z) = 0

Example: The Harmonic Series

nnn

1

4

1

3

1

2

11

1

1

01

limlim n

un

nn

By grouping the terms in the sum as

2

1

22

1

2

1

1

2

1

1

1

8

1

7

1

6

1

5

1

4

1

3

1

2

11

p

p

pppp

Return to TOC

177


Absolute Convergence of Series of Functions

Given a series of functions:

n

i

in zuzS1

If is convergent the series is called absolutely convergent.

n

i

i zu1

If is convergent but is not, the series is called conditionally convergent.

n

i

i zu1

n

i

i zu1

Uniformly Convergence of Sequences and Series

If for the sequence of functions {un(z)} we can find for each ε>0 a number N (ε)

such that for all zR we say that {un} uniformlyconverges to U (z). ( N is a function only of ε and not of z)

NnzUzu n

If the series of functions {Sn(z)} converges to S (z) for all zR we define the remainder

1

:nz

inn zuzSzSzR

The series of functions {Sn(z)} is uniformly convergent to S (z)

if for all for all ε>0 and for all zR we can find a number N (ε)

such that NnzSzS n

x

y C

R

Return to TOC

178

SOLO

z

zofzerosn

n

z

z

z

n

sin

,2,11sin

12

2

zofzerosn

enz

ez

zte

n

n

zz

zt

1

,2,1

1

1

1

0

1

Euler’s Product

2/12/2/

1

2

010

12ln

0

1

2/2/

211

12

11

z

e

zz

e

n

n

z

zeroszTrivialzerosztrivialNon

z

zofpole

z

primep

z

zz

en

ze

z

z

epz

Weierstrass Product

Hadamard Product

1

22 sin1

1

n

zz

n

zz

zz

010

2/12ln

12/112

Re

ss

es

ss

es

Infinite Products

179

SOLO Infinite Products

In 1735 Euler solved the problem, named “Basel Problem” , posed by Mengoli in 1650, by showing that

6

1

4

1

3

1

2

11

2

12232

n n

122

2

2

2

2

2

2

2

19

14

11sin

k k

xxxx

x

x

He did this by developing an Infinite Product for sin x /x:

The roots of sin x are x =0, ±π, ±2π, ±3π,…. However sin x/x is not a polynomial, but Euler assumed (and check it by numerical computation) that it can be factorized using its roots as

We now that if p (x) and q (x) are two polynomials, then using the roots of the two polynomials we have:

m

n

qqqq

pppp

xxxxxxa

xxxxxxa

xq

xp

21

21

We want to show how to express a general solution for complex function f (x) using the zeros and the poles (finite or infinite) of f (x).

Return to TOC

180


The Mittag-Leffler and Weierstrass Theorems

Magnus Gösta Mittag-Leffler1846 - 1927

Karl Theodor Wilhelm Weierstrass

(1815 – 11897)

We want to answer the following questions:

• Can we find f ϵ M (C) so that f has poles exactly a prescribed sequence {zn} that does not cluster in C, and such that f has prescribed principal parts (residiu) at these poles (this refers to fixing the entire portion of the Laurent Series with negative powers at each pole)?

A positive answer to this question was given by Mittag-Leffler

• Can we find f ϵ H (C) so that f has zeros exactly at a given sequence {zn} ?

A positive answer to this question was given by Weierstrass

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http://en.wikipedia.org/wiki/File:Karl_Weierstrass.jpg

181


Definition 1:We say that the Infinite Product converges, if for any N0 > iN the limit

exists and is nonzero.

If this is satisfied then we can compute

N

Nj iN

N

Nj iN

N

Ni iN

N0000lnlimlnlimlimlnln

We transformed the Infinite Product in an Infinite Series, and we know that a necessary (but not sufficient) condition for an Infinite Series to converge is

1lim0lnlim j

jj

j

For simplicity we will define

0lim1 j

jjj aa

1j j

00lim N

N

Nj jN

182


Lemma 2:Let aj ϵ C be such that |aj| < 1. Let . Then

N

j jN aQ1

1:

Nj j

Nj j a

N

aeQe 112

1

Proof:

Since 1 + |aj| ≤ e |aj|

N

j j

N

a

Q

N eaa 111 1

On the other hand, since ex ≤ 1 + 2 x for 0 ≤ x ≤ 1,

NN

aaa

QaaeeeNj

jNj

jNNj j

2/212/21 122

22

111

11 q.e.d.

Proof: Suppose . Then, by the previous Lemma, QN ≤ eM, for all N. Since Q1 ≤Q2 ≤ …., the sequence of “partial products” {QN} converges.Conversely, if the Infinit Product converges to Q, then Q ≥ 1 andfor all N. Then converges.

Maj j

11

1j ja

QaN

j j ln21

Lemma 3:Let aj ϵ C be such that |aj| < 1. Then converges if and only if converges.

11

j ja

1j ja

q.e.d.

183


Proof: Since the product converges, then |aj| → 1, so that aj ≠ 0 for j ≥ j0. Let assume j0 = 1, and define

11

j ja

N

j jN

N

j jN aQandaP11

1:,1:

Note that for a suitable choice of indexes ajk

N

n

n

k j

N

j jN kaaP

1 1111:

Then 111 11 1

N

N

n

n

k j

N

n

n

k jN QaaPkk

and for N, M > 1, N > M

MN

n

Mj jM

n

Mj j

M

j j

NMN

j

M

j jjMN

QQaQ

aaaaPP

11

11111

1

111 1

Hence, {PN} is a Cauchy Sequence, since {QN} is, and it converges.

Lemma 4:If the infinite product converges, then also

converges. Hence if the series converges, also converges.

11

j ja

11

j ja

11

j ja

1j ja

184



We need to prove that {PN} does not converge to zero. By Lemma 2

2

31

NMj jaN

Mj j ea

for M ≥ j0, and N > M. Then using

2

11

2

31111

N

Mj j

N

Mj j aa

for M ≥ j0, and N > M. Hence 2

11

N

Mj ja

so that

012

111limlim 0

11

j

j j

N

Mj j

M

j jj

Nj

aaaP

q.e.d.

11 NN QP

Lemma 4:If the infinite product converges, then also

converges. Hence if the series converges, also converges.

11

j ja

11

j ja

11

j ja

1j ja

Return to TOC

185


Start with some introductory results:

sf

sfsf

sd

d 'ln

TheoremLet f (s) be entire holomorphic (analytic for all s ϵ C) and f (s) ≠ 0 everywhere. There is an entire function g (s) for which f = eg.

Proof:

Since f (s) ≠ 0 and entire, f’ (s) is also entire, and so is f’(s)/ f (s), therefore

entireissf

sd

d

sf

sfsg

sd

dln

':

and taking g (0)= 1 we obtain sgesf q.e.d.

CorollaryIf f (s) is entire holomorphic (analytic) with finitely many zeros {ai≠0}(with multiplicity) and m zeros at s=0, then there exists an entire g (s) such that

nsgm asessf /1

Proof:

Since is entire with no zeros we can apply the Theorem nm asssf /1/

q.e.d.



(1815 – 11897)


186



DefinitionWe define the Weierstrass Elementary Factors as

,2,11

01,

2

2

nes

nsnsE

n

sss

n

LemmaFor |s| ≤ 1, |1 – E (s,n)| ≤ |s|n+1.

Proof: The case n = 0 is trivial. Let n ≥ 1. Let differentiate E (s,n)

n

sss

nn

sss

nn

sss

n

sss

nn

sss

nnnnn

eseseesssensEsd

d

222212

22222

111,

By developing in a Taylor series

0, 2

2

knk

kk

n

sss

n bsbesnsEsd

dn

0

10

1,,k

kk

k

kk saknsE

sd

dsansE

,2,10

0

1,0

21

0

jjn

ba

aaa

nEa

jnjn

n


(1815 – 11897)

Inspired by the fact that

321

1ln&11

321

1ln ss

ss

es s w have the following


187



DefinitionWe define the Weierstrass Elementary Factors as

,2,11

01,

2

2

nes

nsnsE

n

sss

n

LemmaFor |s| ≤ 1, |1 – E (s,n)| ≤ |s|n+1.


01,1

k

nk

kk asansE

So for |s| ≤ 1

1

0

1

1

1

1

11

1

11

1

11

1

,11

,1

nn

nkk

n

nkk

ns

nk

nk

k

n

nk

nkk

n

nk

kk

snEsasas

sassassansE

q.e.d.

188


The Weierstrass Product

Let {sj} be a sequence of complex numbers such that limj→∞ |sj|=+∞. We may assume that 0 < |s1| ≤ |s2| ≤… Let {pj} be integers. Then the Weierstrass Product defined as

converges uniformly on every set {|s|≤r}, to a Holomorphic Entire function F. The zeros of F are precisely the points {sj} counted with the corresponding multiplicity.

1

1

2

1

1

2

1,j

s

s

ps

s

s

s

jj j

j

jp

jjjjes

sp

s

sE

Proof:Let r > 0 be fixed. Let j0 be such that |sj| > r for j ≥ j0. Thus,

11

1,

jj

p

j

p

jj

j s

r

s

sp

s

sE

By the hypothesis on the pj’s,

00

1

1,jj

p

jjj j

j

j

s

rp

s

sE

By Weierstrass M (Majorant Test) it follows that converges uniformly on {|s| ≤ r}, for any r > 0. Then exist C > 0 such that

01,/

jj jj pssE

C

jj

ps

sEp

s

sE

jj jj

eeeCps

sE

jj

jj jj

0

0

0

1,1,

1,

C

jj

pz

zE

jj jj

jj jj

jj

pz

zE

eepz

zEp

z

zEe

jj

Taylorj

j

0000

1,,

1,1,

189


The Weierstrass Product

Let {sj} be a sequence of complex numbers such that limj→∞ |sj|=+∞. We may assume that 0 < |s1| ≤ |s2| ≤… Let {pj} be integers. Then the Weierstrass Product defined as

converges uniformly on every set {|s|≤r}, to a Holomorphic Entire function F. The zeros of F are precisely the points {sj} counted with the corresponding multiplicity.

1

1

2

1

1

2

1,j

s

s

ps

s

s

s

jj j

j

jp

jjjjes

sp

s

sE

Proof:Let r > 0 be fixed. Let j0 be such that |sj| > r for j ≥ j0. Thus,

11

1,

jj

p

j

p

jj

j s

r

s

sp

s

sE

By the hypothesis on the pj’s,

00

1

1,jj

p

jjj j

j

j

s

rp

s

sE

By Weierstrass M (Majorant Test) it follows that converges uniformly on {|s| ≤ r}, for any r > 0. Then exist C > 0 such that

01,/

jj jj pssE

C

jj

ps

sEp

s

sE

jj jj

eeeCps

sE

jj

jj jj

0

0

0

1,1,

1,

C

jj

pz

zE

jj jj

jj jj

jj

pz

zE

eepz

zEp

z

zEe

jj

Taylorj

j

0000

1,,

1,1,

190



LemmaLet {zj} be a sequence of complex numbers such that limj→∞ |zj|=+∞. Then there exists an entire function F whose zeros are precisely the {z j}, counting multiplicity.This function is

11,:

kjj

k jz

zEssF

Return to TOC

191



Definition: Order of an Entire FunctionAn Entire (analytic for all z ϵ C) Function f is said to be of Order ρ, 0 ≤ ρ ≤ +∞, if

meaning that

rasezf r

rz

Osupinf:

0

CzallforeAzfBAexists zB

:0,inf:

0

Examples:

(1) Polynomials have Order 0. Let N be the degree of p (z)

for all ε > 0 and a suitable constant Cε .

raseCrCazazazp rNnn

01

(2) The exponential ex has order 1, and more generally, have order n

and no smaller power of r would suffice.(3) sin z, cos z, sinh z, cosh z have order 1.(4) exp {exp z} has infinite order.

nnnn rzzr eeee Re

nxe

Return to TOC

192


Mittag-Leffler’s Expansion Theorem

Magnus Gösta Mittag-Leffler1846 - 1927

1

110

n nn

n aazafResfzf

Suppose that the only singularities of f (z) in the z-plane are thesimple poles a1, a2,…, arranged in order of increasing absolutevalues. The respective residues of f are Res { f (a1)}, Res { f (a2)}, …

C

x

y

RN

1ana

CN

Proof:

Assume ξ is not a pole of f (z), then has simple polesat a1, a2,…, and ξ.

z

zf

Residue of at an, n = 1,2,… is z

zf

n

nnaz a

afRes

z

zfaz

n

lim

Residue of at ξ is z

zf

fz

zfz

naz

lim

Let take a circle CN at the origin with a radius RN → ∞

By the Residue Theorem

NNCinn n

n

Ca

afResfdz

z

zf

Assume f (z) is analytic at z = 0, then NN

Cinn n

n

Ca

afResfdz

z

zf0

193


Mittag-Leffler’s Expansion Theorem (continue – 1)

C

x

y

RN

1ana

CN


Let take a circle CN at the origin with a radius RN → ∞

NN

Cinn n

n

Ca

afResfdz

z

zf

i

2

1

NN

Cinn n

n

Ca

afResfdz

z

zf

i0

2

1

NN

N

CC

Cinn nn

n

dzzz

zf

idz

zzzf

i

aaafResff

2

11

2

1

110

Since | z-ξ | ≥ | z | - | ξ |=RN - | ξ | for z on CN, we have if | f(z) | ≤ M

0

2limlim

NN

N

RC

R RR

RMdz

zz

zfN

N

N

0lim

N

N

CR

dzzz

zf

1

110

n nn

n aazafResfzf

Therefore using this result and ξ → z, we obtain

q.e.d.

Return to TOC

194



q.e.d.

Suppose that the only singularities of f (z) in the z-plane are the poles a1, a2,…, arranged in order of increasing absolute values, and having Higher Order then One. The respective residues of f are Res { f (a1)}, Res { f (a2)}, … Suppose that exists a Positive Integer p such that for |z| = RN

|f (z)| < RNp+1

and the poles a1, a2,…, an are all inside the Circle of Radius RN around the origin (|a1|≤ |a2|≤…≤ |an | < RN). Then

p

i jp

j

p

jjjj

ii

j jp

j

p

ip

p

a

z

a

z

aazaf

i

zf

aza

zaff

p

zf

zfzf

1 112

11

11

11Res

!

0

Res0!

0!1

0

Proof: Start with the Integral

Nj

N

Cina jp

j

j

pwpzw

Cp

zaa

af

zww

wf

zww

wf

dwzww

wf

iI

1101

1

ResResRes

2

1

C

x

y

RN

1ana

CN

Return to Infinite Product

195




but

111limRes

ppzwpzw z

zf

zww

wfzw

zww

wf

C

x

y

RN

1ana

CN

i

i

ip

pp

iw

pp

p

p

wpw

wd

wfd

zwwd

d

ipi

p

p

zww

wfw

wd

d

pzww

wf

1

!!

!

!

1lim

!

1limRes

1

00

11

010

1

1 !11

ip

ip

ip

p

zw

ip

zwwd

d

p

iip

i

p

ii

i

ip

ip

wpw

zi

f

wd

wfd

zw

ip

ipi

p

pzww

wf

01

01010

!

0

!1

!!

!

!

1limRes

Therefore

Leibnitz Formula for RepeatedDifferentiation of a Product

196




but

Nj Cina j

pj

jp

iip

i

p zaa

af

zi

f

z

zf1

011

Res

!

0

C

x

y

RN

1ana

CN

Therefore

Nj

N

Cina jp

j

j

pwpzw

Cp

zaa

af

zww

wf

zww

wf

dwzww

wf

iI

1101

1

ResResRes

2

1

0max2

2

1

2

11max

11

N

pN

NCN

N

Rn

RwfCN

pN

N

Cp

wfzRR

Rdw

zww

wf

iI

0

Res

!

0

11

011

j jp

j

jp

iip

i

p zaa

af

zi

f

z

zf

11

1

0

Res

!

0

j jp

j

pj

p

i

ii

aza

zaf

i

zfzf

197



We can see that for p = 0 we get

C

x

y

RN

1ana

CN

11

11Res0

Res0

n nnn

n nn

n

aazaff

aza

zaffzf

We recovered the Mittag-Leffler’s Expansion Theorem

11

1

0

Res

!

0

j jp

j

pj

p

i

ii

aza

zaf

i

zfzf

112

11

111

ResRes

jp

j

p

jjjj

j jp

j

pj

a

z

a

z

aazaf

aza

zaf

q.e.d.


Return to TOC

198



An Integral Function is a function which is Analytic for all finite values of z.For example ez, sin z, cos z are Integral Functions. An Integral Function may be regarded as a generalization of a Polynomial.

Let f (z) be an Integral Function (no Poles) with Simple/Non-simple Zeros at a1, a2,…,an,.., arranged in increasing order (|a1|≤ |a2|≤…≤|an|≤…. ). Suppose that exists a Positive Integer p such that for |z| = RN

|f (z)| < RNp+1

and the Zeros a1, a2,…, an are all inside the Circle of Radius RN around the origin (|a1|≤ |a2|≤…≤ |an | < RN).Then f (z) can be expanded as an Infinite Product (Hadamard):

1

1

1

2

11

1

2

2

111 10

j

a

z

pa

z

a

z

j

zczc pj

p

jjp

p ea

zefzf

C

x

y

RN

1ana

CN

Note:1.The minimum p for which |f(z)|<RN

p+1 is called the Order of f(z)2.If f(z) has no poles or zeros then the previous relation reduces to

1110

pp zczcefzf


Return to TOC

199



Proof:

C

x

y

RN

1ana

CN

Let compute: zf

zfzf

zd

d 1

ln

1limlimRes1

21'11

f

fazf

f

faz

f

f j

az

HopitalLj

azaz

jjj

Define

pii

ff

zdd

c z

i

i

i ,,1,0!1

: 0

1

1

112

01

1 111

jp

j

p

jjj

p

i

ii

a

z

a

z

aazzic

zf

zf

All Zeros of f (z) (a1, a2,…,an,..) are Simple Poles of f(1)(z)/f(z), therefore we can apply the previous result and write:

1

12

1

0

0

1

1 11Res

! jp

j

p

jjjaz

p

i

i

z

i

i

a

z

a

z

aazf

f

i

zf

fzd

d

zf

zf

j


200




C

x

y

RN

1ana

CN

Integrating from 0 to z along a path not passing through any of aj, we obtain

1

1

1

2

2

0

11 1

1

2

1ln

0ln

jp

j

p

jjj

jp

i

ii

a

z

pa

z

a

z

a

azzc

f

zf

The values of the logarithms will depend on the path chosen, but when we take exponentials all the ambiguities disappear,

112

01

1 111

jp

j

p

jjj

p

i

ii

a

z

a

z

aazzic

zf

zf

1

1

1

2

11

1

2

2

0

11

10 j

a

z

pa

z

a

z

j

zc pj

p

jj

p

i

ii

ea

ze

f

zf

q.e.d.

If |f(z)| < RNp+1 it will be true for all q > p. If we choose the ρ = min p for which

the inequality holds, then we obtain the Hadamard’s Factorization .


Return to TOC

201

SOLO Primes

Hadamard Infinite Product Expansion of Zeta Function

Graph showing the Trivial Zeros, the Critical Strip and the Critical Line of ζ (z) zeros.

1

1

1

2

11

1

2

2

111 10

j

a

z

pa

z

a

z

j

zczc pj

p

jjp

p ea

zefzf

Since (z-1) ζ (z) is Analytic and has only Zeros we can use the Hadamard Infinite Product Expansion

Zeta Function ζ (z) has Order p=0

2

1

2

10 1 B

The Zero of the Zeta Function ζ (z) are-Trivial Zeros at z=-2n, n=1,2,…- Nontrivial Zeros ρ on the Critical Zone 0 < Re ρ < 1

1

2

0102/10

21101 1

n

n

z

zofzerostrivial

z

zofzerosnontrivial

zc

fzf

en

ze

zezz

Hadamard Infinite Product Expansion of (z-1) ζ (z) is:

pii

ff

zdd

c z

i

i

i ,,1,0!1

: 0

1

1

12ln2/1

2/2ln2/1

0

0'0:

1

0

1

1

zzzf

zf

fc

2

10&

2

2ln0'

1

1

z

forconverges

nzn

z

202

SOLO Primes

Hadamard Infinite Product Expansion of Zeta Function (continue)

Graph showing the Trivial Zeros, the Critical Strip and the Critical Line of ζ (z) zeros.

1

2

010

12ln

211

21

n

n

z

zofzerostrivial

z

zofzerosnontrivial

z

en

ze

zezz

1

22

21

22/

1

n

n

zz

en

ze

z

z

Hadamard Infinite Product Expansion of (z-1) ζ (z) is:

We found the Weierstrass Expansion for the Gamma Function:

100

2/12ln

12/112

Re

zz

ez

zz

ez

2

122

21

22

1

2

ze

zze

en

zzz

n

n

z

Hadamard (1893) used the Weierstrass product theorem to derive this result. The plot above shows the convergence of the formula along the real axis using the first 100 (red), 500 (yellow), 1000 (green), and 2000 (blue) Riemann zeta function zeros.Return to TOC

bxtxxtxtxaxxtfdttf nnn

n

iiiin

b

a

1121100

01lim

SOLO

Riemann Integral

http://en.wikipedia.org/wiki/Riemann_integral

ix 1ix

it

itf

ax 0 bxn

ii xx 1

b

a

dttf

In Riemann Integral we divide the interval [a,b]in n non-overlapping intervals, that decrease asn increases. The value f (ti) is computed inside theintervals.

bxtxxtxtxa nnn 1121100

The Riemann Integral is not always defined, for example:

irationalex

rationalexxf

3

2

The Riemann Integral of this function is not defined.

Georg Friedrich BernhardRiemann

1826 - 1866

Integration

http://en.wikipedia.org/wiki/Image:Riemann.gif

Integration

SOLO

Thomas Joannes Stieltjes

1856 - 1894

Riemann–Stieltjes integral

Bernhard Riemann1826 - 1866

The Stieltjes integral is a generalization of Riemann integral. Let f (x) and α (x) be] real-valued functions defined in the closed interval [a,b]. Take a partition of the interval

and consider a Riemann sum

bxxxa n 10

iii

n

iiii xxxxf ,1

11

If the sum tends to a fixed number I when max(xi-xi-1)→0 then I is called aStieltjes integral or a Riemann-Stieltjes integral. The Stieltjes integral of fwith respect to α is denoted:

xdxf

df

If f and α have a common point of discontinuity, then the integral doesn’t exist.However, if f is continuous and α’ is Riemann integrable over the specific interval

or sometimes simply

xd

xddxfxdxf

:''

http://en.wikipedia.org/wiki/File:ThomasStieltjes.jpeg

http://en.wikipedia.org/wiki/File:Georg_Friedrich_Bernhard_Riemann.jpeg

my

ky

kyE

myE

1M

2M 01 ME

02 ME xfy

SOLO

Lebesgue Integral

Measure

The mean idea of the Lebesgue integral is the notion of Measure.

Definition 1: E (M) є [a,b] is the regionin x є [a,b], of the function f (x) for which Mxf

Definition 2: µ [E (M)] the measure of E (M) is

0 ME

dxME

We can see that µ [E (M)] is the sum of lengths on x axis for which Mxf

From the Figure above we can see that for jumps M1 and M2 021 MEME

Example: Let find the measure of the rationale numbers, ratio of integers, that are countable

n

mrrrrrr k ,,

4

3,

4

1,

3

2,

3

1,

2

15321 3

Since the rationale numbers are discrete we can choose ε > 0 as small as we want and construct an open interval of length ε/2 centered around r1, an interval of ε/22 centered around r2,.., an interval of ε/2k centered around rk

k

rationalsE222 2

00

rationalsE

Integration

xfyyyyxfyyEyydttfbxa

nnibxa

n

i

iiin

b

a

supinflim 110

0

1

a b

0y1y

1ky

1kyky

1ny

ny 1kyE

1kyE kyE

xfy

irationalex

rationalexxf

3

2

SOLO

Lebesgue Integral

Henri Léon Lebesgue1875 - 1941

A function y = f (x) is said to be measurable if the set of points x at which f (x) < c is measurable for any and all choises of the constant c.

The Lebesgue Integral for a measurable function f (x) is defined as:

Example

30131

0

1110/

irationalsErationalsEirationalsExfE

dxxfdxxfdxxfdxxf

3

2

0 1 x

xf

Irationals

Rationals

For a continuous function the Riemann and Lebesgue integrals give the same results.

Integration

IntegrationSOLO

Lebesgue-Stieltjes integration

Thomas Joannes Stieltjes

1856 - 1894

Henri Léon Lebesgue

1875 - 1941

In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.

Let α (x) a monotonic increasing function of x, and define an interval I =(x1,x2).

Define the nonnegative function

12 xxIU The Lebesgue integral with respect to a measure constructed using U (I) is caled Lebesgue-Stieltjes integral, or sometimes Lebesgue-Radon integral.

Johann Karl August Radon

1887 –1956

http://en.wikipedia.org/wiki/File:ThomasStieltjes.jpeg

http://en.wikipedia.org/wiki/File:Johann_Radon.png

IntegrationSOLO

Darboux Integral Lower (green) and upper (green plus lavender) Darboux sums for four subintervals

Jean-Gaston Darboux

1842 - 1917 In real analysis, a branch of mathematics, the Darboux integral or Darboux sum is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. Darboux integrals have the advantage of being simpler to define than Riemann integrals. Darboux integrals are named after their discoverer, Gaston Darboux.

A partition of an interval [a,b] is a finite sequence of values xi such that bxxxa n 10

Definition

Each interval [xi−1,xi] is called a subinterval of the partition. Let ƒ:[a,b]→R be a bounded function, and let nxxxP ,,, 10 be a partition of [a,b]. Let

xfmxfM

iiiixxx

ixxx

i,, 11

inf:;sup:

The upper Darboux sum of ƒ with respect to P is

n

iiiiPf MxxU

11, :

The lower Darboux sum of ƒ with respect to P is

n

iiiiPf mxxL

11, :

http://en.wikipedia.org/wiki/File:Darboux.svg

http://en.wikipedia.org/wiki/File:Darboux.jpg

IntegrationSOLO

Darboux Integral(continue – 1)

Lower (green) and upper (green plus lavender) Darboux sums for four subintervals

Jean-Gaston Darboux

1842 - 1917The upper Darboux sum of ƒ with respect to P is

n

iiiiPf MxxU

11, :

The lower Darboux sum of ƒ with respect to P is

n

iiiiPf mxxL

11, :

The upper Darboux integral of ƒ is baofpartitionaisPUU Pff ,:inf ,

The lower Darboux integral of ƒ is baofpartitionaisPLL Pff ,:inf ,

If Uƒ = Lƒ, then we say that ƒ is Darboux-integrable and set

ff

b

a

LUdttf the common value of the upper and lower Darboux integrals.

http://en.wikipedia.org/wiki/File:Darboux.svg

http://en.wikipedia.org/wiki/File:Darboux.jpg

IntegrationSOLO

Lebesgue Integration

Henri Léon Lebesgue

1875 - 1941

Illustration of a Riemann integral (blue) and a Lebesgue integral (red)

Riemann Integral A sequence of Riemann sums. The numbers in the upper right are the areas of the grey rectangles. They converge to the integral of the function.

Darboux Integral Lower (green) and upper (green plus lavender) Darboux sums for four subintervals

Jean-Gaston Darboux

1842 - 1917

Bernhard Riemann1826 - 1866

Return to TOC

http://en.wikipedia.org/wiki/File:RandLintegrals.svg

http://upload.wikimedia.org/wikipedia/commons/e/ee/Riemann.gif

211

Prime Number Applications

Primes

RSA is an algorithm for public-key cryptography that is based on the presumed difficulty of factoring large integers, the factoring problem. RSA stands for Ron Rivest, Adi Shamir and Leonard Adleman, who first publicly described the algorithm in 1977. Clifford Cocks, an English mathematician, had developed an equivalent system in 1973, but it wasn't declassified until 1997

A user of RSA creates and then publishes the product of two large prime numbers, along with an auxiliary value, as their public key. The prime factors must be kept secret. Anyone can use the public key to encrypt a message, but with currently published methods, if the public key is large enough, only someone with knowledge of the prime factors can feasibly decode the message.[2] Whether breaking RSA encryption is as hard as factoring is an open question known as the RSA problem

Adi Shamir, Ron Rivest andLeonard Adleman

212

Prime Number Applications

Primes

Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but one can also use OpenSSL to generate and examine a real keypair.1.Choose two distinct big (order of 10 308) prime numbers, p and q2. Compute n = p q n is used as the modulus for both the public and private keys. Its length, usually expressed in bits, is the key length 3.Compute φ (n) = (p-1)x(q-1) giving

4. Choose any number 1 < e < φ (n) that is coprime to (not divide) φ (n) e is released as the public key exponent5.Determine d as d-1= e (mod (φ (n)), i.e., d is the multiplicative inverse of e (modulo φ (n)) This is more clearly stated as solve for d given d.e=1 (modulo φ (n)). d is kept as the private key exponent.

The public key consists of the modulus n and the public (or encryption) exponent e. The private key consists of the modulus n and the private (or decryption) exponent d, which must be kept secret. p, q, and φ(n) must also be kept secret because they can be used to calculate d.

213

Prime Number ApplicationsPrimes

Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but one can also use OpenSSL to generate and examine a real keypair.1.Choose two distinct prime numbers, such as

p = 61 and q = 53 (the small prime numbers is for the example only)2.Compute n = p q giving n = 61x53 = 32333.Compute φ (n) = (p-1)x(q-1) giving φ (3233) = (61-1)x(53-1) = 31204.Choose any number 1 < e < 3120 that is coprime to (not divide) 3120 Let e = 175. Compute d, the modular multiplicative inverse of e (mod φ (n)) 17 x d = 1 (mod (3120) d = 2753 ← 17x2753 = 46801 = 3120 x 15 + 1

The public key is n = 3233, e = 17.

For a padded plaintext message m, the encryption function is c (m) = m17 (mod (3233)

The private key is n = 3233, d = 2753. For an encrypted ciphertext c, the decryption function is c2753 (mod (3233)) m (c) = c2753 (mod (3233))Example: m=65, → c (65)=6517 (mod (3233)=2790, m (2790)=27902753 (mod (3233))=65

Return to TOC

214


Primes

215

SOLO Primes

Mathematics Carl Friedrich Gauss taught here in the 19th century. Bernhard Riemann, Johann Peter Gustav Lejeune Dirichlet and a number of significant mathematicians made their contributions to mathematics here. By 1900, David Hilbert and Felix Klein had attracted mathematicians from around the world to Göttingen, which made Göttingen a world mecca of mathematics at the beginning of the 20th century.



Georg Friedrich Bernhard Riemann

)1826– 1866 (

Johann Peter Gustav Lejeune Dirichlet

)1805– 1859 (

David Hilbert)1862 – 1943 (

Christian Felix Klein (1849 – 1925)

216

SOLO


In the 1930s, the university became a focal point for the Nazi crackdown on "Jewish physics", as represented by the work of Albert Einstein. In what was later called the "great purge" of 1933, academics including Max Born, Victor Goldschmidt, James Franck, Eugene Wigner, Leó Szilárd, Edward Teller, Emmy Noether, and Richard Courant were expelled or fled. Most of them fled Nazi Germany for places like the United States, Canada, the United Kingdom, and Ireland.

Max Born (1882–1970)Nobel Physics

Victor Moritz Goldschmidt(1888 –1947)Mineralogist

James Franck (1882 –1964)

Nobel Physicist

Eugene Paul "E. P." Wigner

(1902 – 1995)Nobel Physicist

Leó Szilárd (1898 –1964)

Physicist

Edward Teller (1908 –2003)

PhysicistHydrogen Bomb

Amalie Emmy Noether

)1882 – 1935,(Mathematician

Richard Courant (1888 –1972) Mathematician

Science

Prime numbers