While this is a strong mathematical treatment of Riemann's Zeta function, the steps between equations are very terse and not intuitively obvious. A little more time could have been spent filling in steps between equations. This is not a book to read but to study. If you have not had graduate level mathematics, choose another book.
Author(s): H. M. Edwards
Series: Pure and Applied Mathematics
Edition: Dover Ed
Publisher: Academic Press
Year: 1974
Language: English
Pages: 326
Contents......Page 4
Preface......Page 8
Acknowledgments......Page 11
1.1 The Historical Context of the Paper......Page 12
1.2 The Euler Product Formula......Page 17
1.3 The Factorial Function......Page 18
1.4 The Function \zeta(s)......Page 20
1.5 Values of \zeta(s)......Page 22
1.6 First Proof of the Functional Equation......Page 23
1.7 Second Proof of the Functional Equation......Page 26
1.8 The Function \xi(s)......Page 27
1.9 The Roots \rho of \xi......Page 29
1.10 The Product Representation of \xi(s)......Page 31
1.11 The Connection between \zeta(s) and Primes......Page 33
1.12 Fourier Inversion......Page 34
1.13 Method for Deriving the Formula for J(x)......Page 36
1.14 The Principal Term of J(x)......Page 37
1.15 The Term Involving the Roots \rho......Page 40
1.16 The Remaining Terms......Page 42
1.17 The Formula for \pi(x)......Page 44
1.18 The Density dJ......Page 47
1.19 Questions Unresolved by Riemann......Page 48
2.1 Introduction......Page 50
2.2 Jensen's Theorem......Page 51
2.3 A Simple Estimate of |\xi(0)|......Page 52
2.5 Convergence of the Product......Page 53
2.6 Rate of Growth of the Quotient......Page 54
2.7 Rate of Growth of Even Entire Functions......Page 56
2.8 The Product Formula for \xi......Page 57
3.1 Introduction......Page 59
3.2 Derivation of von Mangoldt's Formula for \psi(x)......Page 61
3.3 The Basic Integral Formula......Page 65
3.4 The Density of the Roots......Page 67
3.5 Proof of von Mangoldt's Formula for \psi(x)......Page 69
3.6 Riemann's Main Formula......Page 72
3.7 Von Mangoldt's Proof of Riemann's Main Formula......Page 73
3.8 Numerical Evaluation of the Constant......Page 77
4.1 Introduction......Page 79
4.2 Hadamard's Proof That Re \rho < 1 for All \rho......Page 81
4.3 Proof That \psi(x) \tilde x......Page 83
4.4 Proof of the Prime Number Theorem......Page 87
5.1 Introduction......Page 89
5.2 An Improvement of Re \rho < 1......Page 90
5.3 De la Vallée Poussin's Estimate of the Error......Page 92
5.4 Other Formulas for \pi(x)......Page 95
5.5 Error Estimates and the Riemann Hypothesis......Page 99
5.6 A Postscript to de la Vallée Poussin's Proof......Page 102
6.1 Introduction......Page 107
6.2 Euler-Maclaurin Summation......Page 109
6.3 Evaluation of \Pi by Euler-Maclaurin Summation. Stirling's Series......Page 117
6.4 Evaluation of \zeta by Euler-Maclaurin Summation......Page 125
6.5 Techniques for Locating Roots on the Line......Page 130
6.6 Techniques for Computing the Number of Roots in a Given Range......Page 138
6.7 Backlund's Estimate of N(T)......Page 143
6.8 Alternative Evaluation of \zeta'/(0)/\zeta(0)......Page 145
7.1 Introduction......Page 147
7.2 Basic Derivation of the Formula......Page 148
7.3 Estimation of the Integral away from the Saddle Point......Page 152
7.4 First Approximation to the Main Integral......Page 156
7.5 Higher Order Approximations......Page 159
7.6 Sample Computations......Page 166
7.7 Error Estimates......Page 173
7.8 Speculations on the Genesis of the Riemann Hypothesis......Page 175
7.9 The Riemann-Siegel Integral Formula......Page 177
8.1 Introduction......Page 182
8.2 Turing's Method......Page 183
8.3 Lehmer's Phenomenon......Page 186
8.4 Computations of Rosser, Yohe, and Schoenfeld......Page 190
9.1 Introduction......Page 193
9.2 Lindelöf's Estimates and His Hypothesis......Page 194
9.3 The Three Circles Theorem......Page 198
9.4 Backlund's Reformulation of the Lindelöf Hypothesis......Page 199
9.5 The Average Value of S(t) Is Zero......Page 201
9.6 The Bohr-Landau Theorem......Page 204
9.7 The Average of |\zeta(O)|^2......Page 206
9.8 Further Results. Landau's Notation o, O......Page 210
10.1 Invariant Operators on R^+ and Their Transforms......Page 214
10.2 Adjoints and Their Transforms......Page 216
10.3 A Self-Adjoint Operator with Transform \xi(s)......Page 217
10.4 The Functional Equation......Page 220
10.5 2\xi(s)/s(s-1) as a Transform......Page 223
10.6 Fourier Inversion......Page 224
10.7 Parseval's Equation......Page 226
10.8 The Values of \zeta(-n)......Page 227
10.9 Möbius Inversion......Page 228
10.10 Ramanujan's Formula......Page 229
11.1 Hardy's Theorem......Page 237
11.2 There Are at Least KT Zeros on the Line......Page 240
11.3 There Are at Least KT log T Zeros on the Line......Page 248
11.4 Proof of a Lemma......Page 257
12.1 The Riemann Hypothesis and the Growth of M{x)......Page 271
12.2 The Riemann Hypothesis and Farey Series......Page 274
12.3 Denjoy's Probabilistic Interpretation of the Riemann Hypothesis......Page 279
12.5 Transforms with Zeros on the Line......Page 280
12.6 Alternative Proof of the Integral Formula......Page 284
12.7 Tauberian Theorems......Page 289
12.8 Chebyshev's Identity......Page 292
12.9 Selberg's Inequality......Page 295
12.10 Elementary Proof of the Prime Number Theorem......Page 299
12.11 Other Zeta Functions. Weil's Theorem......Page 309
Appendix: On the Number of Primes Less Than a Given Magnitude......Page 310
References......Page 317
Index......Page 322
