Gamma

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