| Preface | p. ix |
| Acknowledgments | p. xiii |
| Riemann's Paper | |
| The Historical Context of the Paper | p. 1 |
| The Euler Product Formula | p. 6 |
| The Factorial Function | p. 7 |
| The Function [zeta](s) | p. 9 |
| Values of [zeta](s) | p. 11 |
| First Proof of the Functional Equation | p. 12 |
| Second Proof of the Functional Equation | p. 15 |
| The Function [xi](s) | p. 16 |
| The Roots [rho] of [xi] | p. 18 |
| The Product Representation of [xi](s) | p. 20 |
| The Connection between [zeta](s) and Primes | p. 22 |
| Fourier Inversion | p. 23 |
| Method for Deriving the Formula for J(x) | p. 25 |
| The Principal Term of J(x) | p. 26 |
| The Term Involving the Roots [rho] | p. 29 |
| The Remaining Terms | p. 31 |
| The Formula for [pi](x) | p. 33 |
| The Density dJ | p. 36 |
| Questions Unresolved by Riemann | p. 37 |
| The Product Formula for [xi] | |
| Introduction | p. 39 |
| Jensen's Theorem | p. 40 |
| A Simple Estimate of [xi](s) | p. 41 |
| The Resulting Estimate of the Roots [rho] | p. 42 |
| Convergence of the Product | p. 42 |
| Rate of Growth of the Quotient | p. 43 |
| Rate of Growth of Even Entire Functions | p. 45 |
| The Product Formula for [xi] | p. 46 |
| Riemann's Main Formula | |
| Introduction | p. 48 |
| Derivation of von Mangoldt's Formula for [psi](x) | p. 50 |
| The Basic Integral Formula | p. 54 |
| The Density of the Roots | p. 56 |
| Proof of von Mangoldt's Formula for [psi](x) | p. 58 |
| Riemann's Main Formula | p. 61 |
| Von Mangoldt's Proof of Riemann's Main Formula | p. 62 |
| Numerical Evaluation of the Constant | p. 66 |
| The Prime Number Theorem | |
| Introduction | p. 68 |
| Hadamard's Proof That Re [rho] [ 1 for All [rho] | p. 70 |
| Proof That [psi](x) [similar] x | p. 72 |
| Proof of the Prime Number Theorem | p. 76 |
| De la Vallee Poussin's Theorem | |
| Introduction | p. 78 |
| An Improvement of Re [rho] [ 1 | p. 79 |
| De la Vallee Poussin's Estimate of the Error | p. 81 |
| Other Formulas for [pi](x) | p. 84 |
| Error Estimates and the Riemann Hypothesis | p. 88 |
| A Postscript to de la Vallee Poussin's Proof | p. 91 |
| Numerical Analysis of the Roots by Euler-Maclaurin Summation | |
| Introduction | p. 96 |
| Euler-Maclaurin Summation | p. 98 |
| Evaluation of II by Euler-Maclaurin Summation. Stirling's Series | p. 106 |
| Evaluation of [zeta] by Euler-Maclaurin Summation | p. 114 |
| Techniques for Locating Roots on the Line | p. 119 |
| Techniques for Computing the Number of Roots in a Given Range | p. 127 |
| Backlund's Estimate of N(T) | p. 132 |
| Alternative Evaluation of [zeta]'(0)/[zeta](0) | p. 134 |
| The Riemann-Siegel Formula | |
| Introduction | p. 136 |
| Basic Derivation of the Formula | p. 137 |
| Estimation of the Integral away from the Saddle Point | p. 141 |
| First Approximation to the Main Integral | p. 145 |
| Higher Order Approximations | p. 148 |
| Sample Computations | p. 155 |
| Error Estimates | p. 162 |
| Speculations on the Genesis of the Riemann Hypothesis | p. 164 |
| The Riemann-Siegel Integral Formula | p. 166 |
| Large-Scale Computations | |
| Introduction | p. 171 |
| Turing's Method | p. 172 |
| Lehmer's Phenomenon | p. 175 |
| Computations of Rosser, Yohe, and Schoenfeld | p. 179 |
| The Growth of Zeta as t [infinity] [right arrow] and the Location of Its Zeros | |
| Introduction | p. 182 |
| Lindelof's Estimates and His Hypothesis | p. 183 |
| The Three Circles Theorem | p. 187 |
| Backlund's Reformulation of the Lindelof Hypothesis | p. 188 |
| The Average Value of S(t) Is Zero | p. 190 |
| The Bohr-Landau Theorem | p. 193 |
| The Average of [zeta](s) [superscript 2] | p. 195 |
| Further Results. Landau's Notation o, O | p. 199 |
| Fourier Analysis | |
| Invariant Operators on R[superscript +] and Their Transforms | p. 203 |
| Adjoints and Their Transforms | p. 205 |
| A Self-Adjoint Operator with Transform [xi](s) | p. 206 |
| The Functional Equation | p. 209 |
| 2[xi](s)/s(s - 1) as a Transform | p. 212 |
| Fourier Inversion | p. 213 |
| Parseval's Equation | p. 215 |
| The Values of [zeta](-n) | p. 216 |
| Mobius Inversion | p. 217 |
| Ramanujan's Formula | p. 218 |
| Zeros on the Line | |
| Hardy's Theorem | p. 226 |
| There Are at Least KT Zeros on the Line | p. 229 |
| There Are at Least KT log T Zeros on the Line | p. 237 |
| Proof of a Lemma | p. 246 |
| Miscellany | |
| The Riemann Hypothesis and the Growth of M(x) | p. 260 |
| The Riemann Hypothesis and Farey Series | p. 263 |
| Denjoy's Probabilistic Interpretation of the Riemann Hypothesis | p. 268 |
| An Interesting False Conjecture | p. 269 |
| Transforms with Zeros on the Line | p. 269 |
| Alternative Proof of the Integral Formula | p. 273 |
| Tauberian Theorems | p. 278 |
| Chebyshev's Identity | p. 281 |
| Selberg's Inequality | p. 284 |
| Elementary Proof of the Prime Number Theorem | p. 288 |
| Other Zeta Functions. Weil's Theorem | p. 298 |
| On the Number of Primes Less Than a Given Magnitude | p. 299 |
| References | p. 306 |
| Index | p. 311 |
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