Riemann's Zeta Function: (Dover Books on MaTHEMA 1.4tics)

Superb study of one of the most influential classics in mathematics examines the landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics.

Table of Contents:
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 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