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# Riemann's Zeta Function

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## Description

Topics include Riemann's main formula, the prime number theorem, de la Vallée Poussin's theorem, numerical analysis of roots by Euler-Maclaurin summation, the Riemann-Siegel formula, largescale computations, Fourier analysis, zeros on the line, the Riemann hypothesis and Farey series, alternative proof of the integral formula, Tauberian theorems, Chebyshev's identity, and other related topics.

This inexpensive edition of Edwards' superb high-level study will be welcomed by students and mathematicians. Mathematically inclined general readers will likewise value this influential classic.

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## Product details

- Paperback | 330 pages
- 146 x 222 x 19mm | 340g
- 28 Mar 2003
- Dover Publications Inc.
- New York, United States
- English
- 3 figures
- 0486417409
- 9780486417400
- 168,237

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## Table of contents

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 xi

2.1 Introduction

2.2 Jensen's Theorem

2.3 A Simple Estimate of absolute value of |xi (s)|

2.4 The Resulting Estimate of the Roots rho

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 xi

Chapter 3. Riemann's Main Formula

3.1 Introduction

3.2 Derivation of von Mangoldt's formula for psi (x)

3.3 The Basic Integral Formula

3.4 The Density of the Roots

3.5 Proof of von Mangoldt's Formula for psi (x)

3.6 Riemann's Main Formula

3.7 Von Mangoldt's Proof of Reimann'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 rho<1 for All rho

4.3 Proof That psi (x) ~ x

4.4 Proof of the Prime Number Theorem

Chapter 5. De la Vallée Poussin's Theorem

5.1 Introduction

5.2 An Improvement of Re rho<1

5.3 De la Vallée Poussin's Estimate of the Error

5.4 Other Formulas for pi (x)

5.5 Error Estimates and the Riemann Hypothesis

5.6 A Postscript to de la Vallée Poussin's Proof

Chapter 6. Numerical Analysis of the Roots by Euler-Maclaurin Summation

6.1 Introduction

6.2 Euler-Maclaurin Summation

6.3 Evaluation of PI by Euler-Maclaurin Summation. Stirling's Series

6.4 Evaluation of zeta 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 zeta'(0)/zeta(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

8.1 Introduction

8.2 Turing's Method

8.3 Lehmer's Phenomenon

8.4 Computations of Rosser, Yohe, and Schoenfeld

Chapter 9. The Growth of Zeta as t --> infinity and the Location of Its Zeros

9.1 Introduction

9.2 Lindelöf's Estimates and His Hypothesis

9.3 The Three Circles Theorem

9.4 Backlund's Reformulation of the Lindelöf Hypothesis

9.5 The Average Value of S(t) Is Zero

9.6 The Bohr-Landau Theorem

9.7 The Average of absolute value |zeta(s)| superscript 2

9.8 Further Results. Landau's Notation o, O

Chapter 10. Fourier Analysis

10.1 Invariant Operators on R superscript + and Their Transforms

10.2 Adjoints and Their Transforms

10.3 A Self-Adjoint Operator with Transform xi (s)

10.4 The Functional Equation

10.5 2 xi (s)/s(s - 1) as a Transform

10.6 Fourier Inversion

10.7 Parseval's Equation

10.8 The Values of zeta (-n)

10.9 Möbius Inversion

10.10 Ramanujan's Formula

Chapter 11. Zeros on the Line

11.1 Hardy's Theorem

11.2 There Are at Least KT Zeros on the Line

11.3 There Are at Least KT log T Zeros on the Line

11.4 Proof of a Lemma

Chapter 12. Miscellany

12.1 The Riemann Hypothesis and the Growth of M(x)

12.2 The Riemann Hypothesis and Farey Series

12.3 Denjoy's Probabilistic Interpretation of the Riemann Hypothesis

12.4 An Interesting False Conjecture

12.5 Transforms with Zeros on the Line

12.6 Alternative Proof of the Integral Formula

12.7 Tauberian Theorems

12.8 Chebyshev's Identity

12.9 Selberg's Inequality

12.10 Elementary Proof of the Prime Number Theorem

12.11 Other Zeta Functions. Weil's Theorem

Appendix. On the Number of Primes Less Than a Given Magnitude (By Bernhard Riemann)

References; Index

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