Gamma : Exploring Euler's Constant

Among the myriad of constants that appear in mathematics, p, e, andiare the most familiar. Following closely behind is g, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . up to 1/n, minus the natural logarithm of n--the numerical value being 0.5772156. . .. But unlike its more celebrated colleagues p and e, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction. Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!). Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.

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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Gamma : Exploring Euler's Constant

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