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Riemann's Zeta Function

Riemann's Zeta Function

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作者: Harold M. Edwards 著

出版社: Oversea Publishing House

出版时间: 2001-06

版次: 1

ISBN: 9780486417400

定价: 126.10

装帧: 平装

开本: 其他

纸张: 胶版纸

页数: 330页

正文语种: 英语

丛书: Dover Books on Mathematics

分类: 外文古旧书>英文书>工程技术

14人买过

Superbhigh-levelstudyofoneofthemostinfluentialclassicsinmathematicsexamineslandmark1859publicationentitled?OntheNumberofPrimesLessThanaGivenMagnitude,”andtracesdevelopmentsintheoryinspiredbyit.TopicsincludeRiemann'smainformula,theprimenumbertheorem,theRiemann-Siegelformula,large-scalecomputations,Fourieranalysis,andotherrelatedtopics.EnglishtranslationofRiemann'soriginaldocumentappearsintheAppendix. Preface； Acknowledgments

Chapter 1. Riemann‘s Paper

  1.1 The Historical Context of the Paper

  1.2 The Euler Product Formula

  1.3 The Factorial Function

  1.4 The Function zeta (s)

  1.5 Values of zeta (s)

  1.6 First Proof of the Functional Equation

  1.7 Second Proof of the Functional Equation

  1.8 The Function xi (s)

  1.9 The Roots rho of xi

  1.10 The Product Representation of xi (s)

  1.11 The Connection between zeta (s) and Primes

  1.12 Fourier Inversion

  1.13 Method for Deriving the Formula for J(x)

  1.14 The Principal Term of J(x)

  1.15 The Term Involving the Roots rho

  1.16 The Remaining Terms

  1.17 The Formula for pi (x)

  1.18 The Density dJ

  1.19 Questions Unresolved by Riemann

Chapter 2 The Product Formula for ξ

  2.1 Introduction

  2.2 Jensen's Theorem

  2.3 A Simple Estimate of ξ(s)

  2.4 The Resulting Estimate of the Roots

  2.5 Convergence of the Product

  2.6 Rate of Growth of the Quotient

  2.7 Rate of Growth of Even Entire Functions

  2.8 The Product Formula for ξ

Chapter 3 Riemann's Main Formula

  3.1 Introduction

  3.2 Derivation of yon Mangoldt's Formula for ψ(x)

  3.3 The Basic Integral Formula

  3.4 The Density of the Roots

  3.5 Proof of yon Mangoldt's Formula for ψ(x)

  3.6 Riemann's Main Formula

  3.7 Von Mangoldt's Proof of Riemann's Main Formula

  3.8 Numerical Evaluation of the Constant

Chapter 4 The Prime Number Theorem

  4.1 Introduction

  4.2 Hadamard's Proof That Re p
内容简介:
Superbhigh-levelstudyofoneofthemostinfluentialclassicsinmathematicsexamineslandmark1859publicationentitled?OntheNumberofPrimesLessThanaGivenMagnitude,”andtracesdevelopmentsintheoryinspiredbyit.TopicsincludeRiemann'smainformula,theprimenumbertheorem,theRiemann-Siegelformula,large-scalecomputations,Fourieranalysis,andotherrelatedtopics.EnglishtranslationofRiemann'soriginaldocumentappearsintheAppendix.
目录:
Preface； Acknowledgments

Chapter 1. Riemann‘s Paper

  1.1 The Historical Context of the Paper

  1.2 The Euler Product Formula

  1.3 The Factorial Function

  1.4 The Function zeta (s)

  1.5 Values of zeta (s)

  1.6 First Proof of the Functional Equation

  1.7 Second Proof of the Functional Equation

  1.8 The Function xi (s)

  1.9 The Roots rho of xi

  1.10 The Product Representation of xi (s)

  1.11 The Connection between zeta (s) and Primes

  1.12 Fourier Inversion

  1.13 Method for Deriving the Formula for J(x)

  1.14 The Principal Term of J(x)

  1.15 The Term Involving the Roots rho

  1.16 The Remaining Terms

  1.17 The Formula for pi (x)

  1.18 The Density dJ

  1.19 Questions Unresolved by Riemann

Chapter 2 The Product Formula for ξ

  2.1 Introduction

  2.2 Jensen's Theorem

  2.3 A Simple Estimate of ξ(s)

  2.4 The Resulting Estimate of the Roots

  2.5 Convergence of the Product

  2.6 Rate of Growth of the Quotient

  2.7 Rate of Growth of Even Entire Functions

  2.8 The Product Formula for ξ

Chapter 3 Riemann's Main Formula

  3.1 Introduction

  3.2 Derivation of yon Mangoldt's Formula for ψ(x)

  3.3 The Basic Integral Formula

  3.4 The Density of the Roots

  3.5 Proof of yon Mangoldt's Formula for ψ(x)

  3.6 Riemann's Main Formula

  3.7 Von Mangoldt's Proof of Riemann's Main Formula

  3.8 Numerical Evaluation of the Constant

Chapter 4 The Prime Number Theorem

  4.1 Introduction

  4.2 Hadamard's Proof That Re p

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