Havil J. Gamma: Exploring Euler's Constant

Princeton University Press, 2003. — 266 p.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.Foreword
AcknowledgementsThe Logarithmic Cradle
A Mathematical Nightmare—and an Awakening
The Baron’s Wonderful Canon
A Touch of Kepler
A Touch of Euler
Napier’s Other Ideas
The Harmonic Series
The Principle
Generating Function for H_n
Three Surprising Results
Sub-Harmonic Series
A Gentle Start
Harmonic Series of Primes
The Kempner Series
Madelung’s Constants
Zeta Functions
Where n Is a Positive Integer
Where x Is a Real Number
Two Results to End With
Gamma’s Birthplace
Advent
Birth
The Gamma Function
Exotic Definitions…
… Yet Reasonable Definitions
Gamma Meets Gamma
Complement and Beauty
Euler’s Wonderful Identity
The All-Important Formula…
… And a Hint of Its Usefulness
A Promise Fulfilled
What Is Gamma… Exactly?
Gamma Exists
Gamma Is… What Number?
A Surprisingly Good Improvement
The Germ of a Great Idea
Gamma as a Decimal
Bernoulli Numbers
Euler–Maclaurin Summation
Two Examples
The Implications for Gamma
Gamma as a Fraction
A Mystery
A Challenge
An Answer
Three Results
Irrationals
Irrationals
Pell’s Equation Solved
Filling the Gaps
The Harmonic Alternative
Where Is Gamma?
The Alternating Harmonic Series Revisited
In Analysis
In Number Theory
In Conjecture
In Generalization
It’s a Harmonic World
Ways of Means
Geometric Harmony
Musical Harmony
Setting Records
Testing to Destruction
Crossing the Desert
Shuffling Cards
Quicksort
Collecting a Complete Set
A Putnam Prize Question
Maximum Possible Overhang
Worm on a Band
Optimal Choice
It’s a Logarithmic World
A Measure of Uncertainty
Benford’s Law
Continued-Fraction Behaviour
Problems with Primes
Some Hard Questions about Primes
A Modest Start
A Sort of Answer
Picture the Problem
The Sieve of Eratosthenes
Heuristics
A Letter
The Harmonic Approximation
Different—and Yet the Same
There are Really Two Questions, Not Three
Enter Chebychev with Some Good Ideas
Enter Riemann, Followed by Proof(s)
The Riemann Initiative
Counting Primes the Riemann Way
A New Mathematical Tool
Analytic Continuation
Riemann’s Extension of the Zeta Function
Zeta’s Functional Equation
The Zeros of Zeta
The Evaluation of Π(x) and π(x)
Misleading Evidence
The Von Mangoldt Explicit Formula—and How It Is Used to Prove the Prime Number Theorem
The Riemann Hypothesis
Why Is the Riemann Hypothesis Important?
Real Alternatives
A Back Route to Immortality—Partly Closed
Incentives, Old and New
Progress
Appendix A: The Greek Alphabet
Appendix B: Big Oh Notation
Appendix C: Taylor Expansions
C.1 Degree 1
C.2 Degree 2
C.3 Examples
C.4 Convergence
Appendix D: Complex Function Theory
D.1 Complex Differentiation
D.2 Weierstrass Function
D.3 Complex Logarithms
D.4 Complex Integration
D.5 A Useful Inequality
D.6 The Indefinite Integral
D.7 The Seminal Result
D.8 An Astonishing Consequence
D.9 Taylor Expansions—and an Important Consequence
D.10 Laurent Expansions—and Another Important Consequence
D.11 The Calculus of Residues
D.12 Analytic Continuation
Appendix E: Application to the Zeta Function
E.1 Zeta Analytically Continued
E.2 Zeta’s Functional RelationshipName Index
Subject Index