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Tables of Contents for Riemann's Zeta Function
Chapter/Section Title
Page #
Page Count
Preface
ix
Acknowledgments
xiii
Riemann's Paper
The Historical Context of the Paper
1
5
The Euler Product Formula
6
1
The Factorial Function
7
2
The Function ζ(s)
9
2
Values of ζ(s)
11
1
First Proof of the Functional Equation
12
3
Second Proof of the Functional Equation
15
1
The Function ξ(s)
16
2
The Roots ρ of ξ
18
2
The Product Representation of ξ(s)
20
2
The Connection between ζ(s) and Primes
22
1
Fourier Inversion
23
2
Method for Deriving the Formula for J(x)
25
1
The Principal Term of J(x)
26
3
The Term Involving the Roots ρ
29
2
The Remaining Terms
31
2
The Formula for π(x)
33
3
The Density dJ
36
1
Questions Unresolved by Riemann
37
2
The Product Formula for ξ
Introduction
39
1
Jensen's Theorem
40
1
A Simple Estimate of |ξ(s)|
41
1
The Resulting Estimate of the Roots ρ
42
1
Convergence of the Product
42
1
Rate of Growth of the Quotient
43
2
Rate of Growth of Even Entire Functions
45
1
The Product Formula for ξ
46
2
Riemann's Main Formula
Introduction
48
2
Derivation of von Mangoldt's Formula for ψ(x)
50
4
The Basic Integral Formula
54
2
The Density of the Roots
56
2
Proof of von Mangoldt's Formula for ψ(x)
58
3
Riemann's Main Formula
61
1
Von Mangoldt's Proof of Riemann's Main Formula
62
4
Numerical Evaluation of the Constant
66
2
The Prime Number Theorem
Introduction
68
2
Hadamard's Proof That Re ρ < 1 for All ρ
70
2
Proof That ψ(x) ∼ x
72
4
Proof of the Prime Number Theorem
76
2
De la Vallee Poussin's Theorem
Introduction
78
1
An Improvement of Re ρ < 1
79
2
De la Vallee Poussin's Estimate of the Error
81
3
Other Formulas for π(x)
84
4
Error Estimates and the Riemann Hypothesis
88
3
A Postscript to de la Vallee Poussin's Proof
91
5
Numerical Analysis of the Roots by Euler--Maclaurin Summation
Introduction
96
2
Euler--Maclaurin Summation
98
8
Evaluation of Π by Euler--Maclaurin Summation. Stirling's Series
106
8
Evaluation of ζ by Euler-Maclaurin Summation
114
5
Techniques for Locating Roots on the Line
119
8
Techniques for Computing the Number of Roots in a Given Range
127
5
Backlund's Estimate of N(T)
132
2
Alternative Evaluation of ζ'(O)/ζ(O)
134
2
The Riemann--Siegel Formula
Introduction
136
1
Basic Derivation of the Formula
137
4
Estimation of the Integral away from the Saddle Point
141
4
First Approximation to the Main Integral
145
3
Higher Order Approximations
148
7
Sample Computations
155
7
Error Estimates
162
2
Speculations on the Genesis of the Riemann Hypothesis
164
2
The Riemann--Siegel Integral Formula
166
5
Large-Scale Computations
Introduction
171
1
Turing's Method
172
3
Lehmer's Phenomenon
175
4
Computations of Rosser, Yohe, and Schoenfeld
179
3
The Growth of Zeta as t → ∞ and the Location of Its Zeros
Introduction
182
1
Lindelof's Estimates and His Hypothesis
183
4
The Three Circles Theorem
187
1
Backlund's Reformulation of the Lindelof Hypothesis
188
2
The Average Value of S(t) Is Zero
190
3
The Bohr--Landau Theorem
193
2
The Average of |ζ(s)|2
195
4
Further Results. Landau's Notation o, O
199
4
Fourier Analysis
Invariant Operators on R+ and Their Transforms
203
2
Adjoints and Their Transforms
205
1
A Self-Adjoint Operator with Transform ξ(s)
206
3
The Functional Equation
209
3
2ξ(s)/s(s --- 1) as a Transform
212
1
Fourier Inversion
213
2
Parseval's Equation
215
1
The Values of ζ(--n)
216
1
Mobius Inversion
217
1
Ramanujan's Formula
218
8
Zeros on the Line
Hardy's Theorem
226
3
There Are at Least KT Zeros on the Line
229
8
There Are at Least KT log T Zeros on the Line
237
9
Proof of a Lemma
246
14
Miscellany
The Riemann Hypothesis and the Growth of M(x)
260
3
The Riemann Hypothesis and Farey Series
263
5
Denjoy's Probabilistic Interpretation of the Riemann Hypothesis
268
1
An Interesting False Conjecture
269
1
Transforms with Zeros on the Line
269
4
Alternative Proof of the Integral Formula
273
5
Tauberian Theorems
278
3
Chebyshev's Identity
281
3
Selberg's Inequality
284
4
Elementary Proof of the Prime Number Theorem
288
10
Other Zeta Functions. Weil's Theorem
298
1
Appendix On the Number of Primes Less Than a Given Magnitude
299
7
Bernhard Riemann
References
306
5
Index
311