COMPLEX ANALYSIS in NUMBER THEORY Anatoly A. Karatsuba Steklov Mathematical Institute Russian Academy of Sciences Moscow, Russia CRC Press Boca Raton Ann Arbor London Tokyo
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 logarithmic 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 (s) 18 3.4 Expressing Chebyshev's function as a sum over the complex zeros of (s) 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 21 3.7 Theorem on the uniqueness of ((s) 23 3.8 Proofs of the simplest theorems on the complex zeros of (s) 24
vi 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 (s) in the neighborhood of the critical line 42 2. The Boundary of Zeros 43 2.1 De la Vallee Poussin theorem 43 2.2 Littlewood's theorem 43 2.3 The relationship between the boundary of zeros and the order of growth of \C(s)\ in the neighborhood of unit line 44 2.4 Vinogradov's method in the theory of (s) and Chudakov's theorems 45 2.5 Vinogradov's theorem 46 3. Approximate Equations of the (s) Function 47 3.1 Partial summation and Euler's summation formula 47 3.2 The simplest approximation of (s) 49 3.3 The approximation of a trigonometric sum by a sum of trigonometric integrals 50 3.4 Asymptotic calculations of a certain class of trigonometric integrals 57 3.5 Approximation of a trigonometric sum by a more concise sum 66 3.6 Approximate equations of the ((s) function... 69 3.7 On trigonometric integrals 73
Vll 4. The Method of Trigonometric Sums in the Theory of the (s) Function 77 4.1 The mean 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 ((s) and its corollaries 98 5. Density Theorems 100 5.1 Bertrand's postulate and Chebyshev's theorem.. 100 5.2 Hoheisel's method 100 5.3 Density of zeros of (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 \((s)\ in a Critical Strip 122 6.1 The problem of Dirichlet's divisors 123 6.2 Lindelof's hypothesis 124 6.3 Equivalents of Lindelof's hypothesis 125 6.4 The order of growth of C( + «*)l 126 6.5 Vinogradov's method in Dirichlet's multi-dimensional divisor problem 127 6.6 Omega-theorems 130 7. Universal Properties of the ((s) Function 130 7.1 Bohr's theorems 130 7.2 Voronin's theorems 132 7.3 Theorem on the universal character of ((s)... 134 7.4 More on the universal character of C(-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 (s) 138 8.4 Computations connected with (s) 138
viii 8.5 Functions resembling (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 characters 147 1.2 Principal properties of characters 148 2. Dirichlet i-functions and Prime Numbers in Arithmetic Progressions 149 2.1 Definition of i-functions 149 2.2 The functions v(x;k,i) and tl>(x;k,l) 150 2.3 Dirichlet's theorem on primes 150 3. Zeros of X-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 i-functions and the Number of Classes of Binary Quadratic Forms 154 4.1 Binary quadratic forms and the number of classes. 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. //-Functions and Nonresidues 163 6.1 The concept of a nonresidue 163 6.2 Vinogradov's hypothesis 163
IX 6.3 Lindelof's generalized hypothesis and a nonresidue 164 6.4 The zeros of the Z-functions 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