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| Foreword | |
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| Acknowledgements | |
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| Introduction | |
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| The Logarithmic Cradle | |
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| A Mathematical Nightmare- and an Awakening | |
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| The Baron''s Wonderful Canon | |
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| A Touch of Kepler | |
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| A Touch of Euler | |
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| Napier''s Other Ideas | |
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| The Harmonic Series | |
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| The Principle | |
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| Generating Function for Hn | |
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| Three Surprising Results | |
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| Sub-Harmonic Series | |
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| A Gentle Start | |
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| Harmonic Series of Primes | |
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| The Kempner Series | |
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| Madelung''s Constants | |
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| Zeta Functions | |
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| Where n Is a Positive Integer | |
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| Where x Is a Real Number | |
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| Two Results to End With | |
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| Gamma''s Birthplace | |
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| Advent | |
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| Birth | |
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| The Gamma Function | |
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| Exotic Definitions | |
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| Yet Reasonable Definitions | |
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| Gamma Meets Gamma | |
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| Complement and Beauty | |
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| Euler''s Wonderful Identity | |
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| The All-Important Formula | |
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| And a Hint of Its Usefulness | |
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| A Promise Fulfilled | |
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| What Is Gamma Exactly? | |
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| Gamma Exists | |
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| Gamma Is What Number? | |
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| A Surprisingly Good Improvement | |
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| The Germ of a Great Idea | |
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| Gamma as a Decimal | |
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| Bernoulli Numbers | |
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| Euler -Maclaurin Summation | |
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| Two Examples | |
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| The Implications for Gamma | |
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| Gamma as a Fraction | |
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| A Mystery | |
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| A Challenge | |
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| An Answer | |
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| Three Results | |
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| Irrationals | |
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| Pell''s Equation Solved | |
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| Filling the Gaps | |
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| The Harmonic Alternative | |
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| Where Is Gamma? | |
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| The Alternating Harmonic Series Revisited | |
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| In Analysis | |
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| In Number Theory | |
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| In Conjecture | |
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| In Generalization | |
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| It''s a Harmonic World | |
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| Ways of Means | |
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| Geometric Harmony | |
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| Musical Harmony | |
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| Setting Records | |
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| Testing to Destruction | |
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| Crossing the Desert | |
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| Shuffiing Cards | |
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| Quicksort | |
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| Collecting a Complete Set | |
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| A Putnam Prize Question | |
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| Maximum Possible Overhang | |
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| Worm on a Band | |
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| Optimal Choice | |
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| It''s a Logarithmic World | |
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| A Measure of Uncertainty | |
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| Benford''s Law | |
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| Continued-Fraction Behaviour | |
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| Problems with Primes | |
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| Some Hard Questions about Primes | |
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| A Modest Start | |
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| A Sort of Answer | |
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| Picture the Problem | |
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| The Sieve of Eratosthenes | |
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| Heuristics | |
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| A Letter | |
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| The Harmonic Approximation | |
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| Different-and Yet the Same | |
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| There are Really Two Questions, Not Three | |
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| Enter Chebychev with Some Good Ideas | |
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| Enter Riemann, Followed by Proof(s) | |
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| The Riemann Initiative | |
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| Counting Primes the Riemann Way | |
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| A New Mathematical Tool | |
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| Analytic Continuation | |
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| Riemann''s Extension of the Zeta Function | |
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| Zeta''s Functional Equation | |
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| The Zeros of Zeta | |
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| The Evaluation of (x) and p(x) | |
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| Misleading Evidence | |
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| The Von Mangoldt Explicit Formula-and How It Is Used to Prove the Prime Number Theorem | |
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| The Riemann Hypothesis | |
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| Why Is the Riemann Hypothesis Important? | |
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| Real Alternatives | |
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| A Back Route to Immortality-Partly Closed | |
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| Incentives, Old and New | |
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| Progress | |
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| The Greek Alphabet | |
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| Big Oh Notation | |
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| Taylor Expansions | |
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| Degree 1 | |
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| Degree 2 | |
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| Examples | |
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| Convergence | |
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| Complex Function Theory | |
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| Complex Differentiation | |
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| Weierstrass Function | |
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| Complex Logarithms | |
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| Complex Integration | |
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| A Useful Inequality | |
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| The Indefinite Integral | |
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| The Seminal Result | |
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| An Astonishing Consequence | |
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| Taylor Expansions-and an Important Consequence | |
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| Laurent Expansions-and Another Important Consequence | |
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| The Calculus of Residues | |
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| Analytic Continuation | |
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| Application to the Zeta Function | |
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| Zeta Analytically Continued | |
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| Zeta''s Functional Relationship | |
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| References | |
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| Name Index | |
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| Subject Index | |