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 EulerMaclaurin Summation 

6.1 
Introduction 

6.2 
EulerMaclaurin Summation 

6.3 
Evaluation of PI by EulerMaclaurin Summation. Stirling's Series 

6.4 
Evaluation of zeta by EulerMaclaurin 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 RiemannSiegel 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 RiemannSiegel Integral Formula 
Chapter 8. 
LargeScale 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 BohrLandau 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 SelfAdjoint 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 

