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Riemann Zeta-Function Theory and Applications

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ISBN-10: 0486428133

ISBN-13: 9780486428130

Edition: 2003

Authors: Aleksandar Ivic

List price: $29.95
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Description:

"A thorough and easily accessible account."--MathSciNet, Mathematical Reviews on the Web, American Mathematical Society. This extensive survey presents a comprehensive and coherent account of Riemann zeta-function theory and applications. Starting with elementary theory, it examines exponential integrals and exponential sums, the Voronoi summation formula, the approximate functional equation, the fourth power moment, the zero-free region, mean value estimates over short intervals, higher power moments, and omega results. Additional topics include zeros on the critical line, zero-density estimates, the distribution of primes, the Dirichlet divisor problem and various other divisor problems, and Atkinson's formula for the mean square. End-of-chapter notes supply the history of each chapter's topic and allude to related results not covered by the book. 1985 ed.
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Book details

List price: $29.95
Copyright year: 2003
Publisher: Dover Publications, Incorporated
Publication date: 6/16/2003
Binding: Paperback
Pages: 562
Size: 6.00" wide x 8.75" long x 1.00" tall
Weight: 1.540
Language: English

Notation
Errata
Elementary Theory
Definition of [xi](s) and Elementary Properties
The Functional Equation
The Hadamard Product Formula
The Riemann-von Mangoldt Formula
An Approximate Functional Equation
Mean Value Theorems
Various Dirichlet Series Connected with [xi](s)
Other Zeta-Functions
Unproved Hypotheses
Exponential Integrals and Exponential Sums
Exponential Integrals
Exponential Sums
The Theory of Exponent Pairs
Two-Dimensional Exponent Pairs
The Voronoi Summation Formula
Introduction
The Truncated Voronoi Formula
The Weighted Voronoi Formula
Other Formulas of the Voronoi Type
The Approximate Functional Equations
The Approximate Functional Equation for [xi](s)
The Approximate Functional Equation for [xi superscript 2](s)
The Approximate Functional Equation for Higher Powers
The Reflection Principle
The Fourth Power Moment
Introduction
The Mean Value Theorem for Dirichlet Polynomials
Proof of the Fourth Power Moment Estimate
The Zero-free Region
A Survey of Results
The Method of Vinogradov-Korobov
Estimation of the Zeta Sum
The Order Estimate of [xi](s) Near [sigma] = 1
The Deduction of the Zero-Free Region
Mean Value Estimates Over Short Intervals
Introduction
An Auxiliary Estimate
The Mean Square When [sigma] Is in the Critical Strip
The Mean Square When [sigma] = 1/2
The Order of [xi](s) in the Critical Strip
Third and Fourth Power Moments in Short Intervals
Higher Power Moments
Introduction
Some Convexity Estimates
Power Moments for [sigma] = 1/2
Power Moments for 1/2 [less than sign sigma less than sign] 1
Asymptotic Formulas for Power Moments When 1/2 [less than sign sigma less than sign] 1
Omega Results
Introduction
Omega Results When [sigma greater than or equal] 1
Lemmas on Certain Order Results
Omega Results for 1/2 [less than or equal sigma less than or equal] 1
Lower Bounds for Power Moments When [sigma] = 1/2
Zeros on the Critical Line
Levinson's Method
Zeros on the Critical Line in Short Intervals
Consecutive Zeros on the Critical Line
Zero-Density Estimates
Introduction
The Zero-Detection Method
The Ingham-Huxley Estimates
Estimates for [sigma] Near Unity
Reflection Principle Estimates
Double Zeta Sums
Zero-Density Estimates for 3/4 [less than sign sigma less than sign] 1
Zero-Density Estimates for [sigma] Close to 3/4
The Distribution of Frimes
General Remarks
The Explicit Formula for [psi](x)
The Prime Number Theorem
The Generalized von Mangoldt Function and the Mobius Function
Von Mangoldt's Function in Short Intervals
The Difference between Consecutive Primes
Almost Primes in Short Intervals
Sums of Differences between Consecutive Primes
The Dirichlet Divisor Problem
Introduction
Estimates for [Delta subscript 2](x) and [Delta subscript 3](x)
Estimates of [Delta subscript k](x) by Power Moments of the Zeta-Function
Estimates of [Delta subscript k](x) When k Is Very Large
Estimates of [beta subscript k]
Mean-Square Estimates of [Delta subscript k](x)
Large Values and Power Moments of [Delta subscript k](x)
The Circle Problem
Various Other Divisor Problems
Summatory Functions of Arithmetical Convolutions
Some Applications of the Convolution Method
Three-Dimensional Divisor Problems
Powerful Numbers
Nonisomorphic Abelian Groups of a Given Order
The General Divisor Function d[subscript z](n)
Small Additive Functions
Atkinson's Formula for the Mean Square
Introduction
Proof of Atkinson's Formula
Modified Atkinson's Formula
The Mean Square of E(t)
The Connection Between E(T) and [Delta](x)
Large Values and Power Moments of E(T)
Appendix
References
Author Index
Subject Index