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Foreword | p. xv |
Acknowledgements | p. xvii |
Introduction | p. xix |
The Logarithmic Cradle | p. 1 |
A Mathematical Nightmare- and an Awakening | p. 1 |
The Baron's Wonderful Canon | p. 4 |
A Touch of Kepler | p. 11 |
A Touch of Euler | p. 13 |
Napier's Other Ideas | p. 16 |
The Harmonic Series | p. 21 |
The Principle | p. 21 |
Generating Function for Hn | p. 21 |
Three Surprising Results | p. 22 |
Sub-Harmonic Series | p. 27 |
A Gentle Start | p. 27 |
Harmonic Series of Primes | p. 28 |
The Kempner Series | p. 31 |
Madelung's Constants | p. 33 |
Zeta Functions | p. 37 |
Where n Is a Positive Integer | p. 37 |
Where x Is a Real Number | p. 42 |
Two Results to End With | p. 44 |
Gamma's Birthplace | p. 47 |
Advent | p. 47 |
Birth | p. 49 |
The Gamma Function | p. 53 |
Exotic Definitions | p. 53 |
Yet Reasonable Definitions | p. 56 |
Gamma Meets Gamma | p. 57 |
Complement and Beauty | p. 58 |
Euler's Wonderful Identity | p. 61 |
The All-Important Formula | p. 61 |
And a Hint of Its Usefulness | p. 62 |
A Promise Fulfilled | p. 65 |
What Is Gamma Exactly? | p. 69 |
Gamma Exists | p. 69 |
Gamma Is What Number? | p. 73 |
A Surprisingly Good Improvement | p. 75 |
The Germ of a Great Idea | p. 78 |
Gamma as a Decimal | p. 81 |
Bernoulli Numbers | p. 81 |
Euler -Maclaurin Summation | p. 85 |
Two Examples | p. 86 |
The Implications for Gamma | p. 88 |
Gamma as a Fraction | p. 91 |
A Mystery | p. 91 |
A Challenge | p. 91 |
An Answer | p. 93 |
Three Results | p. 95 |
Irrationals | p. 95 |
Pell's Equation Solved | p. 97 |
Filling the Gaps | p. 98 |
The Harmonic Alternative | p. 98 |
Where Is Gamma? | p. 101 |
The Alternating Harmonic Series Revisited | p. 101 |
In Analysis | p. 105 |
In Number Theory | p. 112 |
In Conjecture | p. 116 |
In Generalization | p. 116 |
It's a Harmonic World | p. 119 |
Ways of Means | p. 119 |
Geometric Harmony | p. 121 |
Musical Harmony | p. 123 |
Setting Records | p. 125 |
Testing to Destruction | p. 126 |
Crossing the Desert | p. 127 |
Shuffiing Cards | p. 127 |
Quicksort | p. 128 |
Collecting a Complete Set | p. 130 |
A Putnam Prize Question | p. 131 |
Maximum Possible Overhang | p. 132 |
Worm on a Band | p. 133 |
Optimal Choice | p. 134 |
It's a Logarithmic World | p. 139 |
A Measure of Uncertainty | p. 139 |
Benford's Law | p. 145 |
Continued-Fraction Behaviour | p. 155 |
Problems with Primes | p. 163 |
Some Hard Questions about Primes | p. 163 |
A Modest Start | p. 164 |
A Sort of Answer | p. 167 |
Picture the Problem | p. 169 |
The Sieve of Eratosthenes | p. 171 |
Heuristics | p. 172 |
A Letter | p. 174 |
The Harmonic Approximation | p. 179 |
Different-and Yet the Same | p. 180 |
There are Really Two Questions, Not Three | p. 182 |
Enter Chebychev with Some Good Ideas | p. 183 |
Enter Riemann, Followed by Proof(s)186 | |
The Riemann Initiative | p. 189 |
Counting Primes the Riemann Way | p. 189 |
A New Mathematical Tool | p. 191 |
Analytic Continuation | p. 191 |
Riemann's Extension of the Zeta Function | p. 193 |
Zeta's Functional Equation | p. 193 |
The Zeros of Zeta | p. 193 |
The Evaluation of (x) and p(x)196 | |
Misleading Evidence | p. 197 |
The Von Mangoldt Explicit Formula-and How It Is Used to Prove the Prime Number Theorem | p. 200 |
The Riemann Hypothesis | p. 202 |
Why Is the Riemann Hypothesis Important? | p. 204 |
Real Alternatives | p. 206 |
A Back Route to Immortality-Partly Closed | p. 207 |
Incentives, Old and New | p. 210 |
Progress | p. 213 |
The Greek Alphabet | p. 217 |
Big Oh Notation | p. 219 |
Taylor Expansions | p. 221 |
Degree 1 | p. 221 |
Degree 2 | p. 221 |
Examples | p. 223 |
Convergence | p. 223 |
Complex Function Theory | p. 225 |
Complex Differentiation | p. 225 |
Weierstrass Function | p. 230 |
Complex Logarithms | p. 231 |
Complex Integration | p. 232 |
A Useful Inequality | p. 235 |
The Indefinite Integral | p. 235 |
The Seminal Result | p. 237 |
An Astonishing Consequence | p. 238 |
Taylor Expansions-and an Important Consequence | p. 239 |
Laurent Expansions-and Another Important Consequence | p. 242 |
The Calculus of Residues | p. 245 |
Analytic Continuation | p. 247 |
Application to the Zeta Function | p. 249 |
Zeta Analytically Continued | p. 249 |
Zeta's Functional Relationship | p. 253 |
References | p. 255 |
Name Index | p. 259 |
Subject Index | p. 263 |
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