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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

AuthorTable of Contents

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

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