Table of Contents
Preface xi
1 Euler's Problem
1.1 Introducing Euler 1
1.2 The Harmonic Series and the Riemann Zeta Function 5
1.3 Euler's Constant, the Zeta Function, and Primes 14
1.4 Euler's Gamma Function, the Reflection Formula, and the Zeta Function 25
1.5 Ramanujan's Master Theorem 37
1.6 Integral Forms for the Harmonic Series and Euler's Constant 43
1.7 Euler's Constant and the Zeta Function Redux (and the Digamma Function, Too) 55
1.8 Calculating ζ(3) 70
2 More Wizard Math and the Zeta Function ζ(s)
2.1 Euler's Infinite Series for ζ(2) 75
2.2 The Beta Function and the Duplication Formula 85
2.3 Euler Almost Computes ζ(3) 94
2.4 Integral Forms of ζ(2) and ζ(3) 97
2.5 Zeta Near s = 1 118
2.6 Zeta Prime at s = 0 126
2.7 Interlude 130
3 Periodic Functions, Fourier Series, and the Zeta Function
3.1 The Concept of a Function 141
3.2 Periodic Functions and Their Fourier Series 149
3.3 Complex Fourier Series and Parseval's Power Formula 157
3.4 Calculating ζ(2n) with Fourier Series 169
3.5 How Fourier Series Fail to Compute ζ(3) 178
3.6 Fourier Transforms and Poisson Summation 184
3.7 The Functional Equation of the Zeta Function 195
4 Euler Sums, the Harmonic Series, and the Zeta Function
4.1 Euler's Original Sums 217
4.2 The Algebra of Euler Sums 220
4.3 Euler's Double Sums 233
4.4 Euler Sums after Euler 238
Epilogue 261
Appendix 1 Solving the Impossible by Changing the Rules 267
Appendix 2 Evaluating $$$ and $$$ 273
Appendix 3 Proof That $$$ Equals Zero 281
