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Among the many constants that appear in mathematics, ð, 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 ð 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.

First published in 2003.

Cover:

Paperback

Edition Number:

1

ISBN:

9780691178103

Pages:

296

Author:

Havil Julian

Publisher:

PRINCETON UNIVERSITY PRESS

Release Year:

2017

Julian Havil is a retired former master at Winchester College, England, where he taught mathematics for thirty-three years. He received a Ph.D. in mathematics from Oxford University. Freeman Dyson is professor emeritus of physics at the Institute for Advanced Study in Princeton. He is the author of several books, including Disturbing the Universe and Origins of Life.

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 Immorta