Riemann's Zeta Function

Cover......Page 1
Title page......Page 3
Date-line......Page 4
Contents......Page 5
PREFACE......Page 9
ACKNOWLEDGMENTS......Page 13
1.1 The Historical Context of the Paper......Page 15
1.2 The Euler Product Formula......Page 20
1.3 The Factorial Function......Page 21
1.4 The Function $\\zeta(s)$......Page 23
1.5 Values of $\\zeta(s)$......Page 25
1.6 First Proof of the Functional Equation......Page 26
1.7 Second Proof of the Functional Equation......Page 29
1.8 The Function $\\xi(s)$......Page 30
1.9 The Roots $\\rho$ of $\\xi$......Page 32
1.10 The Product Representation of $\\xi(s)$......Page 34
1.11 The Connection between $\\zeta(s)$ and Primes......Page 36
1.12 Fourier Inversion......Page 37
1.13 Method for Deriving the Formula for $J(x)$......Page 39
1.14 The Principal Term of $J(x)$......Page 40
1.15 The Term Involving the Roots $\\rho$......Page 43
1.16 The Remaining Terms......Page 45
1.17 The Formula for $\\pi(x)$......Page 47
1.18 The Density $dJ$......Page 50
1.19 Questions Unresolved by Riemann......Page 51
2.1 Introduction......Page 53
2.2 Jensen\'s Theorem......Page 54
2.3 A Simple Estimate of $|\\xi(s)|$ ......Page 55
2.5 Convergence of the Product......Page 56
2.6 Rate of Growth of the Quotient......Page 57
2.7 Rate of Growth of Even Entire Functions......Page 59
2.8 The Product Formula for $\\xi$......Page 60
3.1 Introduction......Page 62
3.2 Derivation of von Mangoldt\'s Formula for $\\psi(x)$......Page 64
3.3 The Basic Integral Formula......Page 68
3.4 The Density of the Roots......Page 70
3.5 Proof of von Mangoldt\'s Formula for $\\psi(x)$......Page 72
3.6 Riemann\'s Main Formula......Page 75
3.7 Von Mangoldt\'s Proof of Riemann\'s Main Formula......Page 76
3.8 Numerical Evaluation of the Constant......Page 80
4.1 Introduction......Page 82
4.2 Hadamard\'s Proof That Re $\\rho < 1$ for All $\\rho$......Page 84
4.3 Proof That $\\psi(x) \\sim x$......Page 86
4.4 Proof of the Prime Number Theorem......Page 90
5.1 Introduction......Page 92
5.2 An Improvement of Re $\\rho < 1$......Page 93
5.3 De la Vallee Poussin\'s Estimate of the Error......Page 95
5.4 Other Formulas for $\\pi(x)$......Page 98
5.5 Error Estimates and the Riemann Hypothesis......Page 102
5.6 A Postscript to de la Vallee Poussin\'s Proof......Page 105
6.1 Introduction......Page 110
6.2 Euler-Maclaurin Summation......Page 112
6.3 Evaluation of $\\Pi$ by Euler-Maclaurin Summation. Stirling\'s Series......Page 120
6.4 Evaluation of $\\zeta$ by Euler-Maclaurin Summation......Page 128
6.5 Techniques for Locating Roots on the Line......Page 133
6.6 Techniques for Computing the Number of Roots in a Given Range......Page 141
6.7 Backlund\'s Estimate of $N(T)$......Page 146
6.8 Alternative Evaluation of $\\zeta\'(0)/\\zeta(0)$......Page 148
7.1 Introduction......Page 150
7.2 Basic Derivation of the Formula......Page 151
7.3 Estimation of the Integral away from the Saddle Point......Page 155
7.4 First Approximation to the Main Integral......Page 159
7.5 Higher Order Approximations......Page 162
7.6 Sample Computations......Page 169
7.7 Error Estimates......Page 176
7.8 Speculations on the Genesis of the Riemann Hypothesis......Page 178
7.9 The Kiciuann-Sicgel Integral Formula......Page 180
8.1 Introduction......Page 185
8.2 Turing\'s Method......Page 186
8.3 Lehmer\'s Phenomenon......Page 189
8.4 Computations of Rosser, Yohe, and Schoenfeld......Page 193
9.1 Introduction......Page 196
9.2 Lindeloef\'s Estimates and His Hypothesis......Page 197
9.3 The Three Circles Theorem......Page 201
9.4 Backlund\'s Reformulation of the Lindeloef Hypothesis......Page 202
9.5 The Average Value of $S(t)$ Is Zero......Page 204
9.6 The Bohr-Landau Theorem......Page 207
9.7 The Average of $|\\zeta(s)|^2$......Page 209
9.8 Further Results. Landau\'s Notation $o$, $O$......Page 213
10.1 Invariant Operators on $R^+$ and Their Transforms......Page 217
10.2 Adjoints and Their Transforms......Page 219
10.3 A Self-Adjoint Operator with Transform $\\xi(s)$......Page 220
10.4 The Functional Equation......Page 223
10.5 $2\\xi(s)/s(s-1)$ as a Transform......Page 226
10.6 Fourier Inversion......Page 227
10.7 Parseval\'s Equation......Page 229
10.8 The Values of $\\zeta(-n)$......Page 230
10.9 Moebius Inversion......Page 231
10.10 Ramanujan\'s Formula......Page 232
11.1 Hardy\'s Theorem......Page 240
11.2 There Are at Least $KT$ Zeros on the Line......Page 243
11.3 There Are at Least $KT \\log T$ Zeros on the Line......Page 251
11.4 Proof of a Lemma......Page 260
12.1 The Riemann Hypothesis and the Growth of $M(x)$......Page 274
12.2 The Riemann Hypothesis and Farey Series......Page 277
12.3 Denjoy\'s Probabilistic Interpretation of the Riemann Hypothesis......Page 282
12.5 Transforms with Zeros on the Line......Page 283
12.6 Alternative Proof of the Integral Formula......Page 287
12.7 Tauberian Theorems......Page 292
12.8 Chebyshev\'s Identity......Page 295
12.9 Selberg\'s Inequality......Page 298
12.10 Elementary Proof of the Prime Number Theorem......Page 302
12.11 Other Zeta Functions. Weil\'s Theorem......Page 312
APPENDIX On the Number of Primes Less Than a Given Magnitude (By Bernhard Riemann)......Page 313
REFERENCES......Page 320
INDEX......Page 325