CONTENTS
Introduction = 1
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 = 8
1.2 Sum functions = 11
2. Summation Formula = 13
2.1 Perron's formula = 13
2.2 Expressing Chebyshev's function in terms of the integral of the logarithnlic derivative of Riemann's zeta-function = 14
3. Riemann's Zeta-Function and Its Simplest Properties = 15
3.1 The functional equation = 15
3.2 Riemann's hypotheses = 17
3.3 The simplest theorems on the zeros of ζ(s) = 18
3.4 Expressing Chebyshev's function as a sum over the complex zeros of ζ(s) = 19
3.5 The asymptotic law of distribution of prime numbErs = 20
3.6 Riemaiin's hypothesis concerning the complex zeros of ζ(s) and the problem of the theory of prime numbers = 21
3.7 Theorem on the uniqueness of ζ(s) = 23
3.8 Proofs of the simplest theorems on the complex zeros of ζ(s) = 24
Chapter 2. The Theory of Riemann's Zeta-Function = 31
1. Zeros on the Critical Line = 31
1.1 Hardy's theorem = 31
1.2 Theorems of Hardy and Littlewood = 31
1.3 Hardy's function and Hardy's method = 32
1.4 Titchmarsh's discrete method = 35
1.5 Selberg's theorem = 35
1.6 Estimates of Selberg's constant = 36
1.7 Moser's theorems = 37
1.8 Selberg's hypothesis = 38
1.9 Zeros of the derivatives of Hardy's function = 39
1.10 The latest results = 40
1.11 Distribution of zeros in the mean = 41
1.12 Density of zeros on the critical line = 41
1.13 The zeros of ζ(s) in the neighborhood of the critical line = 42
2. The Boundary of Zeros = 43
2.1 De la Valke Poussin theorem = 43
2.2 Littlewood's theorem = 43
2.3 The relationship between the boundary of zeros and the order of growth of │ζ(s)│in the neighborhood of unit line = 44
2.4 Vinogradov's method in the theory of ζ(s) and Chudakov's theorems = 45
2.5 Vinogradov's theorem = 46
3. Approximate Equations of the ζ(s) Function = 47
3.1 Partial summation and Euler's summation formula = 47
3.2 The simplest approximation of ζ(s) = 49
3.3 The approximation of a trigonometric sum by a sum of trigonometric integrals = 50
3.4 Asymptotic calculations of a certain dass of trigonometric integrals = 57
3.5 Approximation of a trigonometric sum by a more concise sum = 66
3.6 Approximate equations of the ζ(s) function = 69
3.7 On trigonometric integrals = 73
4. The Method of Trigonometric Sums in the Theory of the ζ(s) Function = 77
4.1 The meaa value. of the degree of the modulus of a trigonometric sum = 77
4.2 Simple lemmas = 78
4.3 The basic recurrent inequality = 83
4.4 Vinogradov's mean-value theorem = 89
4.5 The estimate of the zeta sum and its consequences = 91
4.6 The current boundary of zeros of ζ(s) and its corollaries = 98
5. Density TheoTems = 100
5.1 Bertrand's postulate and Chebyshev's theorem = 100
5.2 Hoheisel's method = 100
5.3 Density of zeros of ζ(s) = 102
5.4 Density theorems = 103
5.5 Proof of Huxley's density theorem = 104
5.6 Three problems of the number theory solvable by Hoheisel's method = 120
6. The Order of Growth of │ζ(s)│ in a Critical Strip = 122
6.1 The problem of Dirichlet's di'visors = 123
6.2 Lindel6f's hypothesis = 124
6.3 Equivalents of Lindel6f's hypothesis = 125
6.4 The order of growth of │ζ($${1 \over 2}$$+ it )│= 126
6.5 Vinogradov's method, in Dirichlet's multi-dimensional divisor problem = 127
6.6 Omega-theorems = 130
7. Universal Properties of the ζ(s) Function = 130
7.1 Bohr's theorems = 130
7.2 Voronin's theorems = 132
7.3 Theorem on the universal character of ζ(s) = 134
7.4 More on the universal character of ζ(s) = 135
8. Riemann's Hypothesis, Its Equivalents, Computations = 135
8.1 Mertens' hypothesis = 136
8.2 Turan's hypothesis and its refutation = 137
8.3 A billion and a half complex zeros of ζ(s) = 138
8.4 Computations connected with ζ(s) = 138
8.5 Functions resembling ζ(s) but having complex zeros on the right of the critical line = 139
8.6 Epstein's zeta-functions = 140
8.7 A new approach to the problem of zeros, lying on the critical line, of some Dirichlet series = 141
Chapter 3. Dirichlet L-Functions = 147
1. Dirichlet's Characters = 147
1.1 Definition of character's = 147
1.2 Principal properties of characters = 148
2. Dirichlet L-Functions and Prime Numbers in Arithmetic Progressions = 149
2.1 Definition of L-functions = 149
2.2 The functions π(x; k, 1) and ψ(x; k, 1) = 150
2.3 Dirichlet's theorem on primes = 150
3. Zeros of L-Functions = 152
3.1 The boundary of zeros. Page's theorems = 152
3.2 Siegel's theorem = 153
3.3 Zeros on the critical line = 153
4. Real Zeros of L-Functions and the Number of Classes of Binary Quadratic Forms = 154
4.1 Binary quadratic forms and the number of dasses = 154
4.2 Dirichlet's formulas = 156
4.3 Gauss' problem and Siegel's theorem = 156
4.4 Prime numbers in arithmetic progressions = 157
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 numbers based on the ideas of Hardy-Littlewood-Linnik = 160
6. L-Functions and Nonresidues = 163
6.1 The concept of a nonresidue = 163
6.2 Vinogradov's hypothesis = 163
6.3 Lindel$$\ddot o$$f's generalized hypothesis and a nonresidue = 164
6.4 The zeros of the L-function: and nonresidues = 164
7. Approximate Equations = 165
7.1 Stating the problem = 165
7.2 Lavrik's general theorem = 165
8. On Primitive Roots = 168
8.1 The concept of a primitive root = 168
8.2 Artin's hypothesis = 168
8.3 Hooley's conditional theorem = 168
References = 170
Author Index = 183
Subject Index = 185
