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.

The Complex Integration Method and Its Application in 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