Complex Analysis in Number Theory

2018·7.91 MB·english

by  Anatoly A. Karatsuba

Il
Preface
The application of the methods of mathematical analysis and, in
Library of Congress Cataloging-in-Publication Data particular, of the theory of functions of a complex variable to the
solution of problems of the number theory contributed to a consider-
Karatsuba, Anatolii Alekseevich.
Complex analysis in number theory/by Anatoly A. Karatsuba able progress in this branch of mathematics. Here is what I.M. Vino-
p.
cm. gradov said in this connection in his review entitled "On the Problems
Includes bibliographical references and index.
of the Analytic Number Theory" published in the proceedings of the
rsBN 0-8493-2866-7
l. Number theory. 2. Functions of complex variables. November jubilee session of the USSR Academy of Sciences dedicated
3. Mathematical analysis. I. Title. to the 15th anniversary of the USSR: "Analysis makes it possible to
Q4241.K32t 1995
512' .13-rJc20 94-34262 extend considerably the range of problems of the number theory and
CIP provides for a more rapid development of this science. I also want
to point out one more useful feature of the analytic method in the
number theory. While solving new difficult problems, analysis itself
develops and gets more perfect. Dirichlet's series and the theory of
This book contains inforrnation obtained from authentic and highly regarded sources. Reprinted the ((s) function can serve as examples as well as some properties of
rnaterial is q0oted with permission, and sources are indicateti. A wide variety of references are listed.
Reasonable efforts have becn made to publish reliable data and information, but the author and the Bessel's functions, a number of remarkable theorems relating to the
publisher cannot assurne responsibility for the validity of all materials or for the consequences of their theory of functions of a complex variable (for instance, the theorems
use.
of Lindelóf, Phragmen, Mellin), discontinuous sums and integrals etc.
Neithcr this book nor any part may be reproduced or transmitted in any form or by any means,
electronic or mechanical, including photocopying, microfilming, and recording, or by any information Thus, the application of the analytic method to the number theory
storage or retrieval system, without prior permission in writing frorn the publisher. enriches this science with new valuable achievements and, at the same
CRC Press, Inc.'s consent does not extend to copying for general distribution, for promotion, fbr
time, develops and perfects the analysis itself."
crcating new works, or for resale. Specific perniission must be obtained in writing from CRC Press for
such copying. This mono$aph primarily deals with the application of analysis
Direct all inquirics to CRC Press, Inc.,2000 Corporate Blvd., N.W., Boca Raton, Florida 33431.
to the problems of the theory of prime numbers. At the same time,
O 199.5 by CRC Press, Inc. it presents the results that appeared in the framework of the number
theory but actually refer to analysis. The monograph does not cover
No clairn to original U.S. Governrncnt works
Intemational Standard Book Number 0-8493-2866-'7 the whole wealth of material connected with the indicated problems
Libruy of Congress Card Nunber 94-34262 of the number theory. The same refers to the literature cited. The
Printcdin(hcunitedStatesof America | 2 3 4 -5 6 7 8 9 0
Printetl on acid-free oaner
lv Preface
content of the monogîaph was discussed with G.I. Arkhipov, S'M'
voronin and V.N. Chubarikov and their useful remarks considerably
perfected the exposition. I am very grateful to them. I want to ex-
press my deep gratitude to my wife, D.V. Senchenko, for her constant
support and attention to mY work.
Contents
Introduction
.
Chapter 1. The Complex Integration Method and Its
Application in Number Theory 8
1. Generating Functions in Number Theory 8
1.1
Dirichlet's series B
1.2 functions
Sum 11
2. Formula
Summation 13
2.1, Perron's formula 13
2.2 Expressing Chebyshev's function in terms of the
integral of the logarithmic derivative of Riemann's
zeta-function
14
3. Riemann's Zeta-Punction and Its Simplest Properties 15
3.1 The functional equation 15
3.2 hypotheses
Riemann's 17
3.3 The simplest theorems on the zeros of ((") . . . . . 18
3.4 Expressing Chebyshev's function as a sum over the
((r) . .
complex zeros of 19
3.5 The asymptotic law of distribution of prime numbers 20
3.6 Riemann's hypothesis concerning the complex
zeros of ((s) and the problem of the theory of prime
numbers 2l
3.7 Theorem on the uniqueness of ((s) . . . . . 23
3.8 Proofs of the simplest theorems on the complex
zerosof(("). .....24
lv Preface
content of the monogîaph was discussed with G.I. Arkhipov, S'M'
voronin and V.N. Chubarikov and their useful remarks considerably
perfected the exposition. I am very grateful to them. I want to ex-
press my deep gratitude to my wife, D.V. Senchenko, for her constant
support and attention to mY work.
Contents
Introduction
.
Chapter 1. The Complex Integration Method and Its
Application in Number Theory 8
1. Generating Functions in Number Theory 8
1.1
Dirichlet's series B
1.2 functions
Sum 11
2. Formula
Summation 13
2.1, Perron's formula 13
2.2 Expressing Chebyshev's function in terms of the
integral of the logarithmic derivative of Riemann's
zeta-function
14
3. Riemann's Zeta-Punction and Its Simplest Properties 15
3.1 The functional equation 15
3.2 hypotheses
Riemann's 17
3.3 The simplest theorems on the zeros of ((") . . . . . 18
3.4 Expressing Chebyshev's function as a sum over the
((r) . .
complex zeros of 19
3.5 The asymptotic law of distribution of prime numbers 20
3.6 Riemann's hypothesis concerning the complex
zeros of ((s) and the problem of the theory of prime
numbers 2l
3.7 Theorem on the uniqueness of ((s) . . . . . 23
3.8 Proofs of the simplest theorems on the complex
zerosof(("). .....24
vl Contents Contents vll
Chapter 2. The Theory of Riemann's Zeta-Function 3t_ 4. The Method of Trigonometric Sums in the Theory of the
1. Zeros on the Critical Line 31 ((s) Function 77
1.1 Hardy's theorem 31 4.1 The mean value of the degree of the modulus of a
7.2
Theorems of Hardy and Littlewood 31 trigonometric sum 77
1.3 Hardy's function and Hardy's method 32 4.2 Simple lemmas 78
1..4 Titchmarsh's discrete method 35 4.3 The basic recurrent inequality 83
1.5 Selberg's theorem 35 4.4 Vinogradov's mean-value theorem 89
1.6 Bstimates of Selberg's constant 36 4.5 The estimate of the zeta sum and its consequences 91
1.7 Moser's theorems 37 4.6 The current boundary of zeros of ((s) and its corol-
1.8
Selberg's hypothesis 38 laries 98
1.9
Zeros of the derivatives of Hardy's function 39 5. Density Theorems 100
1.10 The latest results 40 5.1 Bertrand's postulate and Chebyshev's theorem 100
1.11 Distribution of zeros in the rnean . 4l 5.2 Hoheisel's method 100
I.l2 Density of zeros on the critical line 4I 5.3 Density of zeros of ((s) t02
1.13 The zeros of ((s) in the neighborhood of the critical 5.4 Density theorems 103
line . 42 5.5 Proof of Huxley's density theorem r04
2. The Boundary of Zeros 43 5.6 Three problems of the number theory solvable by
2.r De la Vallèe Poussin theorem 43 Hoheisel's method 120
2.2 Littlewood's theorem 43 6. The Order of Growth of l((s)l in a Critical Strip 122
2.3 The relationship between the boundary of zeros 6.1 The problem of Dirichlet's divisors t23
and the order of growth of l((s)l in the neighbor- 6.2 Lindelóf's hypothesis t24
hood of unit line 44 6.3 Equivalents of Lindelóf 's hypothesis 125
2.4 Vinogradov's method in the theory of ((s) and Chu- 6.4 The order of growth of l((à + it)l . 126
dakov's theorems 45 6.5 Vinogradov's method in Dirichlet's multi-dimen-
2.5
Vinogradov'stheorem 46 sional divisor problem r27
3. Approximate Equations of the ((s) Function 47 6.6 Omega-theorems 130
3.1 Partial summation and Euler's summation formula 47 7. Universal Properties of the ((s) Function 130
3.2 The simplest approximation of ((") . 49 7.1 Bohr's theorems 130
3.3 The approximation of a trigonometric sum by a 7.2 Voronin's theorems r32
sum of trigonometric integrals 50 7.3 Theorem on the universal character of ((s) 134
3.4 Asymptotic calculations of a certain class of trigono- 7.4 More on the universal character of ((s) 135
metric integrals DI 8. Riemann's Hypothesis, Its Equivalents, Computations . 135
3.5 Approximation of a trigonometric sum by a more 8.1 Mertens'hypothesis 136
concise sum 66 8.2 Turan's hypothesis and its refutation t37
3.6 Approximate equations of the ((s) function 69 8.3 A billion and a half complex zeros of ((s) 138
3.7 On trigonometric integrals 73 8.4 Computations connected with ((.s) 138
vl Contents Contents vll
Chapter 2. The Theory of Riemann's Zeta-Function 3t_ 4. The Method of Trigonometric Sums in the Theory of the
1. Zeros on the Critical Line 31 ((s) Function 77
1.1 Hardy's theorem 31 4.1 The mean value of the degree of the modulus of a
7.2
Theorems of Hardy and Littlewood 31 trigonometric sum 77
1.3 Hardy's function and Hardy's method 32 4.2 Simple lemmas 78
1..4 Titchmarsh's discrete method 35 4.3 The basic recurrent inequality 83
1.5 Selberg's theorem 35 4.4 Vinogradov's mean-value theorem 89
1.6 Bstimates of Selberg's constant 36 4.5 The estimate of the zeta sum and its consequences 91
1.7 Moser's theorems 37 4.6 The current boundary of zeros of ((s) and its corol-
1.8
Selberg's hypothesis 38 laries 98
1.9
Zeros of the derivatives of Hardy's function 39 5. Density Theorems 100
1.10 The latest results 40 5.1 Bertrand's postulate and Chebyshev's theorem 100
1.11 Distribution of zeros in the rnean . 4l 5.2 Hoheisel's method 100
I.l2 Density of zeros on the critical line 4I 5.3 Density of zeros of ((s) t02
1.13 The zeros of ((s) in the neighborhood of the critical 5.4 Density theorems 103
line . 42 5.5 Proof of Huxley's density theorem r04
2. The Boundary of Zeros 43 5.6 Three problems of the number theory solvable by
2.r De la Vallèe Poussin theorem 43 Hoheisel's method 120
2.2 Littlewood's theorem 43 6. The Order of Growth of l((s)l in a Critical Strip 122
2.3 The relationship between the boundary of zeros 6.1 The problem of Dirichlet's divisors t23
and the order of growth of l((s)l in the neighbor- 6.2 Lindelóf's hypothesis t24
hood of unit line 44 6.3 Equivalents of Lindelóf 's hypothesis 125
2.4 Vinogradov's method in the theory of ((s) and Chu- 6.4 The order of growth of l((à + it)l . 126
dakov's theorems 45 6.5 Vinogradov's method in Dirichlet's multi-dimen-
2.5
Vinogradov'stheorem 46 sional divisor problem r27
3. Approximate Equations of the ((s) Function 47 6.6 Omega-theorems 130
3.1 Partial summation and Euler's summation formula 47 7. Universal Properties of the ((s) Function 130
3.2 The simplest approximation of ((") . 49 7.1 Bohr's theorems 130
3.3 The approximation of a trigonometric sum by a 7.2 Voronin's theorems r32
sum of trigonometric integrals 50 7.3 Theorem on the universal character of ((s) 134
3.4 Asymptotic calculations of a certain class of trigono- 7.4 More on the universal character of ((s) 135
metric integrals DI 8. Riemann's Hypothesis, Its Equivalents, Computations . 135
3.5 Approximation of a trigonometric sum by a more 8.1 Mertens'hypothesis 136
concise sum 66 8.2 Turan's hypothesis and its refutation t37
3.6 Approximate equations of the ((s) function 69 8.3 A billion and a half complex zeros of ((s) 138
3.7 On trigonometric integrals 73 8.4 Computations connected with ((.s) 138
vttl Contents Contents tx
8.5 Functions resembling ((s) but having complex 6.3 Lindelóf's generalized hypothesis and a nonresidue 164
zeros on the right of the critical line 139 6.4 The zeros of the -t-functions and nonresidues 764
8.6 Epstein'szeta-functions 140 7. Approximate Equations 165
8.7 A new approach to the problem of zeros, lying on 7.1 Stating the problem. 165
the critical line, of some Dirichlet series t4l 7.2 Lavrik's general theorem i65
8. OnPrimitiveRoots ..... i68
Chapter 3. Dirichlet L-Functions L47
1. Dirichlet's Characters r47 8.1 The concept of a primitive root 168
11,..12 DPreinfinciiptioanl porof pcehratrieasct eorfs c.h a.racters t14478 88..23 AHortoinle'sy' sh cyopnoditthioenasl itsh eorem 116688
References i70
2. Dirichlet .t-Functions and Prime Numbers in Arithmetic
Authorlndex .....
183
Progressions t49
Subjectlndex .....185
2.1
Definition of .t-functions 149
2.2 The functions z'(r;k,l) and ,b(*;k,r) . . . 150
2.3 Dirichlet's theorem on primes 150
3. Zeros of -t-Functions . . r52
3.1 The boundary of zeros. Page's theorems 152
3.2
Siegel's theorem 153
3.3 Zeros on the critical line 153
aA. Real Zeros of tr-Functions and the Number of Classes of
Binary Quadratic Forms r54
4.1 Binary quadratic forms and the number of classes r54
.
4.2
Dirichlet's formulas 156
4.3
Gauss'problem and Siegel's theorem 156
4.4 Prime numbers in arithmetic progressions r57
5. Density Theorems 158
5.1 Linnik's density theorems 158
5.2
Density theorems of a large sieve and the Bombieri-
Vinogradov theorem 158
5.3
Current density theorems 160
5.4 Proof of Vinogradov's theorem on three prime num-
bers based on the ideas of Hardy-Littlewood-Linnik 160
6. ,t-Functions and Nonresidues 163
6.1 The concept of a nonresidue 163
6.2
Vinogradov's hypothesis 163
vttl Contents Contents tx
8.5 Functions resembling ((s) but having complex 6.3 Lindelóf's generalized hypothesis and a nonresidue 164
zeros on the right of the critical line 139 6.4 The zeros of the -t-functions and nonresidues 764
8.6 Epstein'szeta-functions 140 7. Approximate Equations 165
8.7 A new approach to the problem of zeros, lying on 7.1 Stating the problem. 165
the critical line, of some Dirichlet series t4l 7.2 Lavrik's general theorem i65
8. OnPrimitiveRoots ..... i68
Chapter 3. Dirichlet L-Functions L47
1. Dirichlet's Characters r47 8.1 The concept of a primitive root 168
11,..12 DPreinfinciiptioanl porof pcehratrieasct eorfs c.h a.racters t14478 88..23 AHortoinle'sy' sh cyopnoditthioenasl itsh eorem 116688
References i70
2. Dirichlet .t-Functions and Prime Numbers in Arithmetic
Authorlndex .....
183
Progressions t49
Subjectlndex .....185
2.1
Definition of .t-functions 149
2.2 The functions z'(r;k,l) and ,b(*;k,r) . . . 150
2.3 Dirichlet's theorem on primes 150
3. Zeros of -t-Functions . . r52
3.1 The boundary of zeros. Page's theorems 152
3.2
Siegel's theorem 153
3.3 Zeros on the critical line 153
aA. Real Zeros of tr-Functions and the Number of Classes of
Binary Quadratic Forms r54
4.1 Binary quadratic forms and the number of classes r54
.
4.2
Dirichlet's formulas 156
4.3
Gauss'problem and Siegel's theorem 156
4.4 Prime numbers in arithmetic progressions r57
5. Density Theorems 158
5.1 Linnik's density theorems 158
5.2
Density theorems of a large sieve and the Bombieri-
Vinogradov theorem 158
5.3
Current density theorems 160
5.4 Proof of Vinogradov's theorem on three prime num-
bers based on the ideas of Hardy-Littlewood-Linnik 160
6. ,t-Functions and Nonresidues 163
6.1 The concept of a nonresidue 163
6.2
Vinogradov's hypothesis 163
fntroduction
The number theory studies the properties of integers. The concept
of integers refers not only to numbers of the natural scale l, 2, 3,. . .
(positive integers) but also to zeto and negative integers -7, -2,
-3,. ... The sum, the difference and the product of two integers
are also integers. But the quotient of the division of one number by
another may be an integer and may be not. As a rule, in what follows
I denote by letters only integers. And if a noninteger is denoted by
a letter, then it is either clear from the context or will be specially
specified. When we use the letter q to denote the quotient of a by ó,
we have î = q, i.e. a = bq. In such a case we say that a is divisible
by ó or that ò divides a. The number b is a divisor of the number
o and the number o is a multiple of the number ó. Every natural
number a, which exceeds unity, has two obvious divisors, namely, 1
and a. If besides these divisors a has one more divisor, ó, 1 ( b 1 o,
then a is a composite number, otherwise a is a prime number, or
simply a prirne. For instance,2,3,5,7,lI ,13 are prime numbers
whereas the number 6 is composite since the numbers 1,2,3,6 are
its divisors. Thus the scale of natural numbers, except for 1, falls
in two sets, namely, prime numbers and composite numbers. When
multiplying prime numbers together, we obtain composite numbers
and, conversely, when isolating prime divisors of the number a, we
represent a as the product of prime factors, i.e. a = pt...pr. It turns
out that euery natural number can be represented as the product of
primes and this representation is unique with an accuracy to within
the order of factors. The last statement has long been regarded as an
obvious fact and only Gauss proved it as a theorem. This theorem is