COMPLEX ANALYSIS in NUMBER THEORY

Transcription

1 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

2 Introduction 1 Chapter 1. The Complex Integration Method and Its Application in Number Theory 8 1. Generating Functions in Number Theory Dirichlet's series Sum functions Summation Formula Perron's formula Expressing Chebyshev's function in terms of the integral of the logarithmic derivative of Riemann's zeta-function Riemann's Zeta-Function and Its Simplest Properties The functional equation Riemann's hypotheses The simplest theorems on the zeros of (s) Expressing Chebyshev's function as a sum over the complex zeros of (s) The asymptotic law of distribution of prime numbers Riemann's hypothesis concerning the complex zeros of ((s) and the problem of the theory of prime numbers Theorem on the uniqueness of ((s) Proofs of the simplest theorems on the complex zeros of (s) 24

3 vi Chapter 2. The Theory of Riemann's Zeta-Function Zeros on the Critical Line Hardy's theorem Theorems of Hardy and Littlewood Hardy's function and Hardy's method Titchmarsh's discrete method Selberg's theorem Estimates of Selberg's constant Moser's theorems Selberg's hypothesis Zeros of the derivatives of Hardy's function The latest results Distribution of zeros in the mean Density of zeros on the critical line The zeros of (s) in the neighborhood of the critical line The Boundary of Zeros De la Vallee Poussin theorem Littlewood's theorem The relationship between the boundary of zeros and the order of growth of \C(s)\ in the neighborhood of unit line Vinogradov's method in the theory of (s) and Chudakov's theorems Vinogradov's theorem Approximate Equations of the (s) Function Partial summation and Euler's summation formula The simplest approximation of (s) The approximation of a trigonometric sum by a sum of trigonometric integrals Asymptotic calculations of a certain class of trigonometric integrals Approximation of a trigonometric sum by a more concise sum Approximate equations of the ((s) function On trigonometric integrals 73

4 Vll 4. The Method of Trigonometric Sums in the Theory of the (s) Function The mean value of the degree of the modulus of a trigonometric sum Simple lemmas The basic recurrent inequality Vinogradov's mean-value theorem The estimate of the zeta sum and its consequences The current boundary of zeros of ((s) and its corollaries Density Theorems Bertrand's postulate and Chebyshev's theorem Hoheisel's method Density of zeros of (s) Density theorems Proof of Huxley's density theorem Three problems of the number theory solvable by Hoheisel's method The Order of Growth of \((s)\ in a Critical Strip The problem of Dirichlet's divisors Lindelof's hypothesis Equivalents of Lindelof's hypothesis The order of growth of C( + «*)l Vinogradov's method in Dirichlet's multi-dimensional divisor problem Omega-theorems Universal Properties of the ((s) Function Bohr's theorems Voronin's theorems Theorem on the universal character of ((s) More on the universal character of C(-s) Riemann's Hypothesis, Its Equivalents, Computations Mertens' hypothesis Turan's hypothesis and its refutation A billion and a half complex zeros of (s) Computations connected with (s) 138

5 viii 8.5 Functions resembling (s) but having complex zeros on the right of the critical line Epstein's zeta-functions A new approach to the problem of zeros, lying on the critical line, of some Dirichlet series 141 Chapter 3. Dirichlet L-Functions Dirichlet's Characters Definition of characters Principal properties of characters Dirichlet i-functions and Prime Numbers in Arithmetic Progressions Definition of i-functions The functions v(x;k,i) and tl>(x;k,l) Dirichlet's theorem on primes Zeros of X-Functions The boundary of zeros. Page's theorems Siegel's theorem Zeros on the critical line Real Zeros of i-functions and the Number of Classes of Binary Quadratic Forms Binary quadratic forms and the number of classes Dirichlet's formulas Gauss' problem and Siegel's theorem Prime numbers in arithmetic progressions Density Theorems Linnik's density theorems Density theorems of a large sieve and the Bombieri- Vinogradov theorem Current density theorems Proof of Vinogradov's theorem on three prime numbers based on the ideas of Hardy-Littlewood-Linnik //-Functions and Nonresidues The concept of a nonresidue Vinogradov's hypothesis 163

6 IX 6.3 Lindelof's generalized hypothesis and a nonresidue The zeros of the Z-functions and nonresidues Approximate Equations Stating the problem Lavrik's general theorem On Primitive Roots The concept of a primitive root Artin's hypothesis Hooley's conditional theorem 168 References 170 Author Index 183 Subject Index 185