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 for Riemann’s Zeta Function

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
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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