In pursuit of zeta-3: the world's most mysterious unsolved math problem

Preface

1. Euler's Problem: 1.1 Introducing Euler

1.2 The harmonic series and the Riemann Zeta function

1.3 Euler's constant, the zeta function, and primes

1.4 Euler's Gamma function, the reflection formula, and the zeta function

1.5 Ramanujan's master theorem

1.6 Integral forms for the harmonic series and Euler's constant

1.7 Euler's constant and the zeta function redux (and the digamma function, too)

1.8 Calculating Z(3)

2. More Wizard Math and the Zeta Function Z(s): 2.1 Euler's infinite series for Z(2)

2.2 The Beta function and the duplication formula

2.3 Euler almost computes Z(3)

2.4 Integral forms of Z(2) and Z(3)

2.5 Zeta near s=1

2.6 Zeta prime at s=0

2.7 Interlude

3. Periodic functions, Fourier series, and the Zeta function: 3.1 The concept of a function

3.2 Periodic functions and their Fourier series

3.3 Complex Fourier sries and Parseval's power formula

3.4 Calculating Z(2n) with Fourier series

3.5 How Fourier series fail to compute Z(3)

3.6 Fourier transforms and Poisson summation

3.7 The functional equation of the zeta function

4. Euler sums, the harmonic series, and the zeta function: 4.1 Euler's original sums

4.2 The algebra of the Euler sums

4.3 Euler's double sums

4.4 Euler sums after Euler

Epilogue

Appendix 1: Solving the impossible by changing the rules

Appendix 2: Evaluating (formula)

Appendix 3: Proof that (formula) equals zero

Appendix 4: Double integration reversal isn't always legal

Appendix 5: Impossibility results from computer science

Challenge: Problem solutions

Acknowledgments

Index