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# Riemann's Zeta Function

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## Description

Bernhard Riemann's eight-page paper entitled "On the Number of Primes Less Than a Given Magnitude" was a landmark publication of 1859 that directly influenced generations of great mathematicians, among them Hadamard, Landau, Hardy, Siegel, Jensen, Bohr, Selberg, Artin, and Hecke. This text, by a noted mathematician and educator, examines and amplifies the paper itself, and traces the developments in theory inspired by it. (An English translation of the original document appears in the Appendix.)

Topics include Riemann's main formula, the prime number theorem, de la Vallee 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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Topics include Riemann's main formula, the prime number theorem, de la Vallee 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.

show more

## Product details

- Paperback | 330 pages
- 134.62 x 210.82 x 17.78mm | 340.19g
- 28 Mar 2003
- Dover Publications Inc.
- New York, United States
- English
- 3 figures
- 0486417409
- 9780486417400
- 180,956

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## Table of contents

Preface; AcknowledgmentsChapter 1. Riemann's Paper1.1 The Historical Context of the Paper1.2 The Euler Product Formula1.3 The Factorial Function1.4 The Function zeta (s)1.5 Values of zeta (s)1.6 First Proof of the Functional Equation1.7 Second Proof of the Functional Equation1.8 The Function xi (s)1.9 The Roots rho of xi1.10 The Product Representation of xi (s)1.11 The Connection between zeta (s) and Primes1.12 Fourier Inversion1.13 Method for Deriving the Formula for J(x)1.14 The Principal Term of J(x)1.15 The Term Involving the Roots rho1.16 The Remaining Terms1.17 The Formula for pi (x)1.18 The Density dJ1.19 Questions Unresolved by RiemannChapter 2. The Product Formula for xi2.1 Introduction2.2 Jensen's Theorem2.3 A Simple Estimate of absolute value of |xi (s)|2.4 The Resulting Estimate of the Roots rho2.5 Convergence of the Product2.6 Rate of Growth of the Quotient2.7 Rate of Growth of Even Entire Functions2.8 The Product Formula for xiChapter 3. Riemann's Main Formula3.1 Introduction3.2 Derivation of von Mangoldt's formula for psi (x)3.3 The Basic Integral Formula3.4 The Density of the Roots3.5 Proof of von Mangoldt's Formula for psi (x)3.6 Riemann's Main Formula3.7 Von Mangoldt's Proof of Reimann's Main Formula3.8 Numerical Evaluation of the ConstantChapter 4. The Prime Number Theorem4.1 Introduction4.2 Hadamard's Proof That Re rho<1 for All rho4.3 Proof That psi (x) ~ x4.4 Proof of the Prime Number TheoremChapter 5. De la Vallée Poussin's Theorem5.1 Introduction5.2 An Improvement of Re rho<15.3 De la Vallée Poussin's Estimate of the Error5.4 Other Formulas for pi (x)5.5 Error Estimates and the Riemann Hypothesis5.6 A Postscript to de la Vallée Poussin's ProofChapter 6. Numerical Analysis of the Roots by Euler-Maclaurin Summation6.1 Introduction6.2 Euler-Maclaurin Summation6.3 Evaluation of PI by Euler-Maclaurin Summation. Stirling's Series6.4 Evaluation of zeta by Euler-Maclaurin Summation6.5 Techniques for Locating Roots on the Line6.6 Techniques for Computing the Number of Roots in a Given Range6.7 Backlund's Estimate of N(T)6.8 Alternative Evaluation of zeta'(0)/zeta(0)Chapter 7. The Riemann-Siegel Formula7.1 Introduction7.2 Basic Derivation of the Formula7.3 Estimation of the Integral away from the Saddle Point7.4 First Approximation to the Main Integral7.5 Higher Order Approximations7.6 Sample Computations7.7 Error Estimates7.8 Speculations on the Genesis of the Riemann Hypothesis7.9 The Riemann-Siegel Integral FormulaChapter 8. Large-Scale Computations8.1 Introduction8.2 Turing's Method8.3 Lehmer's Phenomenon8.4 Computations of Rosser, Yohe, and SchoenfeldChapter 9. The Growth of Zeta as t --> infinity and the Location of Its Zeros9.1 Introduction9.2 Lindelöf's Estimates and His Hypothesis9.3 The Three Circles Theorem9.4 Backlund's Reformulation of the Lindelöf Hypothesis9.5 The Average Value of S(t) Is Zero9.6 The Bohr-Landau Theorem9.7 The Average of absolute value |zeta(s)| superscript 29.8 Further Results. Landau's Notation o, OChapter 10. Fourier Analysis10.1 Invariant Operators on R superscript + and Their Transforms10.2 Adjoints and Their Transforms10.3 A Self-Adjoint Operator with Transform xi (s)10.4 The Functional Equation10.5 2 xi (s)/s(s - 1) as a Transform10.6 Fourier Inversion10.7 Parseval's Equation10.8 The Values of zeta (-n)10.9 Möbius Inversion10.10 Ramanujan's FormulaChapter 11. Zeros on the Line11.1 Hardy's Theorem11.2 There Are at Least KT Zeros on the Line11.3 There Are at Least KT log T Zeros on the Line11.4 Proof of a LemmaChapter 12. Miscellany12.1 The Riemann Hypothesis and the Growth of M(x)12.2 The Riemann Hypothesis and Farey Series12.3 Denjoy's Probabilistic Interpretation of the Riemann Hypothesis12.4 An Interesting False Conjecture12.5 Transforms with Zeros on the Line12.6 Alternative Proof of the Integral Formula12.7 Tauberian Theorems12.8 Chebyshev's Identity12.9 Selberg's Inequality12.10 Elementary Proof of the Prime Number Theorem12.11 Other Zeta Functions. Weil's TheoremAppendix. On the Number of Primes Less Than a Given Magnitude (By Bernhard Riemann)References; Index

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