Riemannis zeta function 黎曼Zeta函数

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<1 for All p
4.3 Proof That ψ(x) ,,- x
4.4 Proof of the Prime Number Theorem
Chapter 5 De la ValiSe Poussin's Theorem
5.1 Introduction
5.2 An Improvement of Re p < 1
5.3 De la Vallee Poussin's Estimate of the Error
5.4 Other Formulas for π(x)
5.5 Error Estimates and the Riemann Hypothesis
5.6 A Postscript to de la Vallee Poussin's Proof
Chapter 6 Nuinerical Analysis of the Roots by Euler-Maclaurin Summation
6.1 Introduction
6.2 Euler-Maclaurin Summation
6.3 Evaluation of II by Euler-Maclaurin Summation. Stirling's Series
6.4 Evaluation of by Euler-Maclaurin Summation
6.5 Techniques for Locating Roots on the Line
6.6 Techniques for Computing the Number of Roots in a Given Range
6.7 Backlund's Estimate of N(T)
6.8 Alternative Evaluation of ζ(0)/ζ(0)
Chapter 7 The Riemann-Siegel Formula
7.1 Introduction
7.2 Basic Derivation of the Formula
7.3 Estimation of the Integral away from the Saddle Point
7.4 First Approximation to the Main Integral
7.5 Higher Order Approximations
7.6 Sample Computations
7.7 Error Estimates
7.8 Speculations on the Genesis of the Riemann Hypothesis
7.9 The Riemann-Siegel Integral Formula
Chapter 8 Large-Scale Computations
Chapter 9 The Growth of Zeta as t-and the Location of Its Zeros
Chapter 10 Fourier Analysis
Chapter 11 Zeros on the Line
Chapter 12 Miscellany
APPENDIX On the Number of Primes Less Than a Given Magnitude
PEFERENCES
INDEX