This book examines the application of complex analysis methods to the theory of prime numbers. In an easy to understand manner, a connection is established between arithmetic problems and those of zero distribution for special functions. Main achievements in this field of mathematics are described. Indicated is a connection between the famous Riemann zeta-function and the structure of the universe, information theory, and quantum mechanics. The theory of Riemann zeta-function and, specifically, distribution of its zeros are presented in a concise and comprehensive way. The full proofs of some modern theorems are given. Significant methods of the analysis are also demonstrated as applied to fundamental problems of number theory.
Generating Functions in Number Theory
Summation Formula
Riemann's Zeta-Function and Its Simplest Properties
The Theory of Riemann's Zeta-Function
Zeros on the Critical Line
The Boundary of Zeros
Approximate Equations of the z(s) Function
The Method of Trigonometric Sums in the Theory of the z(s) Function
Density Theorems
The Order of Growth of |z(s)| in a Critical Strip
Universal Properties of the z(s) Function
Riemann's Hypothesis, Its Equivalents, Computations
Dirichlet L-Functions: Dirichlet's Characters
Dirichlet L-Functions and Prime Numbers in Arithmetic Progressions
Zeros of L-Functions
Real Zeros of L-Functions and the Number of Classes of Binary Quadratic Forms
Density Theorems
L-Functions and Nonresidues
Approximate Equations
On Primitive Roots
References
Author Index
Subject Index
Biography
Karatsuba\, Anatoly A.
