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Among the many constants that appear in mathematics, p, e, and i are the most familiar. Following closely behind is y, 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.
Julian Havil With a foreword by Freeman Dyson
Foreword xv
Acknowledgements xvii
Introduction xix
Chapter One
The Logarithmic Cradle 1
1.1 A Mathematical Nightmare- and an Awakening 1
1.2 The Baron's Wonderful Canon 4
1.3 A Touch of Kepler 11
1.4 A Touch of Euler 13
1.5 Napier's Other Ideas 16
Chapter Two
The Harmonic Series 21
2.1 The Principle 21
2.2 Generating Function for Hn 21
2.3 Three Surprising Results 22
Chapter Three
Sub-Harmonic Series 27
3.1 A Gentle Start 27
3.2 Harmonic Series of Primes 28
3.3 The Kempner Series 31
3.4 Madelung's Constants 33
Chapter Four
Zeta Functions 37
4.1 Where n Is a Positive Integer 37
4.2 Where x Is a Real Number 42
4.3 Two Results to End With 44
Chapter Five
Gamma's Birthplace 47
5.1 Advent 47
5.2 Birth 49
Chapter Six
The Gamma Function 53
6.1 Exotic Definitions 53
6.2 Yet Reasonable Definitions 56
6.3 Gamma Meets Gamma 57
6.4 Complement and Beauty 58
Chapter Seven
Euler's Wonderful Identity 61
7.1 The All-Important Formula 61
7.2 And a Hint of Its Usefulness 62
Chapter Eight
A Promise Fulfilled 65
Chapter Nine
What Is Gamma Exactly? 69
9.1 Gamma Exists 69
9.2 Gamma Is What Number? 73
9.3 A Surprisingly Good Improvement 75
9.4 The Germ of a Great Idea 78
Chapter Ten
Gamma as a Decimal 81
10.1 Bernoulli Numbers 81
10.2 Euler -Maclaurin Summation 85
10.3 Two Examples 86
10.4 The Implications for Gamma 88
Chapter Eleven
Gamma as a Fraction 91
11.1 A Mystery 91
11.2 A Challenge 91
11.3 An Answer 93
11.4 Three Results 95
11.5 Irrationals 95
11.6 Pell's Equation Solved 97
11.7 Filling the Gaps 98
11.8 The Harmonic Alternative 98
Chapter Twelve
Where Is Gamma? 101
12.1 The Alternating Harmonic Series Revisited 101
12.2 In Analysis 105
12.3 In Number Theory 112
12.4 In Conjecture 116
12.5 In Generalization 116
Chapter Thirteen
It's a Harmonic World 119
13.1 Ways of Means 119
13.2 Geometric Harmony 121
13.3 Musical Harmony 123
13.4 Setting Records 125
13.5 Testing to Destruction 126
13.6 Crossing the Desert 127
13.7 Shuffiing Cards 127
13.8 Quicksort 128
13.9 Collecting a Complete Set 130
13.10 A Putnam Prize Question 131
13.11 Maximum Possible Overhang 132
13.12 Worm on a Band 133
13.13 Optimal Choice 134
Chapter Fourteen
It's a Logarithmic World 139
14.1 A Measure of Uncertainty 139
14.2 Benford's Law 145
14.3 Continued-Fraction Behaviour 155
Chapter Fifteen
Problems with Primes 163
15.1 Some Hard Questions about Primes 163
15.2 A Modest Start 164
15.3 A Sort of Answer 167
15.4 Picture the Problem 169
15.5 The Sieve of Eratosthenes 171
15.6 Heuristics 172
15.7 A Letter 174
15.8 The Harmonic Approximation 179
15.9 Different-and Yet the Same 180
15.10 There are Really Two Questions, Not Three 182
15.11 Enter Chebychev with Some Good Ideas 183
15.12 Enter Riemann, Followed by Proof(s)186
Chapter Sixteen
The Riemann Initiative 189
16.1 Counting Primes the Riemann Way 189
16.2 A New Mathematical Tool 191
16.3 Analytic Continuation 191
16.4 Riemann's Extension of the Zeta Function 193
16.5 Zeta's Functional Equation 193
16.6 The Zeros of Zeta 193
16.7 The Evaluation of (x) and p(x)196
16.8 Misleading Evidence 197
16.9 The Von Mangoldt Explicit Formula-and How It Is Used to Prove the Prime Number Theorem 200
16.10 The Riemann Hypothesis 202
16.11 Why Is the Riemann Hypothesis Important? 204
16.12 Real Alternatives 206
16.13 A Back Route to Immortality-Partly Closed 207
16.14 Incentives, Old and New 210
16.15 Progress 213
Appendix A
The Greek Alphabet 217
Appendix B
Big Oh Notation 219
Appendix C
Taylor Expansions 221
C.1 Degree 1 221
C.2 Degree 2 221
C.3 Examples 223
C.4 Convergence 223
Appendix D
Complex Function Theory 225
D.1 Complex Differentiation 225
D.2 Weierstrass Function 230
D.3 Complex Logarithms 231
D.4 Complex Integration 232
D.5 A Useful Inequality 235
D.6 The Indefinite Integral 235
D.7 The Seminal Result 237
D.8 An Astonishing Consequence 238
D.9 Taylor Expansions-and an Important Consequence 239
D.10 Laurent Expansions-and Another Important Consequence 242
D.11 The Calculus of Residues 245
D.12 Analytic Continuation 247
Appendix E
Application to the Zeta Function 249
E.1 Zeta Analytically Continued 249
E.2 Zeta's Functional Relationship 253
References 255
Name Index 259
Subject Index 263
