# Gamma : Exploring Euler's Constant

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## Description

Among the myriad of constants that appear in mathematics, p, e, and i are 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.

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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.

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## Product details

- Hardback | 296 pages
- 152 x 235 x 24.64mm | 567g
- 17 Mar 2003
- Princeton University Press
- New Jersey, United States
- English
- 2 halftones. 87 line illus. 20 tables.
- 0691099839
- 9780691099835
- 983,203

## Flap copy

"I like this book very much. So much, in fact, that I found myself muttering 'neat stuff!' all the way through. While it is about an important topic, there isn't a single competitor. This amazing oversight by past authors is presumably the result of the topic requiring an author with a pretty sophisticated mathematical personality. Havil clearly has that. His skillful weaving of mathematics and history makes the book a 'fun' read. Many instructors will surely find the book attractive."--Paul J. Nahin, author of "Duelling Idiots" and "Other Probability Puzzlers" and "An Imaginary Tale""This is an excellent book, mathematically as well as historically. It represents a significant contribution to the literature on mathematics and its history at the upper undergraduate and graduate levels. Julian Havil injects genuine excitement into the topic."--Eli Maor, author of "e: The Story of a Number"

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## Table of contents

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

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## Review quote

[A] wonderful book... Havil's emphasis on historical context and his conversational style make this a pleasure to read... Gamma is a gold mine of irresistible mathematical nuggets. Anyone with a serious interest in maths will find it richly rewarding. -- Ben Longstaff, New Scientist This book is a joy from start to finish. -- Gerry Leversha, Mathematical Gazette [Gamma] is not a book about mathematics, but a book of mathematics... [It] is something like a picaresque novel; the hero, Euler's constant g, serves as the unifying motif through a wide range of mathematical adventures. -- Dan Segal, Notices of the American Mathematical Society The book is enjoyable for many reasons. Here are just two. First, the explanations are not only complete, but they have the right amount of generality... Second, the pleasure Havil has in contemplating this material is infectious. -- Jeremy Gray, MAA Online It is only fitting that someone should write a book about gamma, or Euler's constant. Havil takes on this task and does an excellent job. -- Choice This book is accessible to a wide range of readers, and should particularly appeal to those who feel a love for mathematics and are dissuaded by the dryness and formality of text-books, but are also not satisfied by the less rigorous approach of most popular books. Mathematics is presented throughout as something connected to reality... Many readers will find in this book exactly what they have been missing. -- Mohammad Akbar, Plus Magazine This book is written in an informal, engaging, and often amusing style. The author takes pains to make the mathematics clear. He writes about the mathematical geniuses of the past with reverence and awe. It is especially nice that the mathematical topics are discussed within a historical context. -- Ward R. Stewart, Mathematics Teacher

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## About Julian Havil

Julian Havil is a Master at Winchester College, England, where he has taught mathematics for nearly thirty years. He received a Ph.D. in mathematics from Oxford University. Freeman Dyson is Professor Emeritus of Physics at the Institute for Advanced Study, Princeton. He is the author of several books, including "Disturbing the Universe" and "Origins of Life".

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