1st Edition

Complex Analysis in Number Theory

By Anatoly A. Karatsuba Copyright 1994
    208 Pages
    by CRC Press

    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
    Author Index
    Subject Index


    Karatsuba\, Anatoly A.