Complex Analysis in Number Theory: 1st Edition (Hardback) book cover

Complex Analysis in Number Theory

1st Edition

By Anatoly A. Karatsuba

CRC Press

208 pages

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Hardback: 9780849328664
pub: 1994-11-22
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Description

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.

Table of Contents

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

Subject Categories

BISAC Subject Codes/Headings:
MAT022000
MATHEMATICS / Number Theory
MAT037000
MATHEMATICS / Functional Analysis