An engrossing look at the history and importance of a centuries-old but still unanswered math problem
For centuries, mathematicians the world over have tried, and failed, to solve the zeta-3 problem. Math genius Leonhard Euler attempted it in the 1700s and came up short. The straightforward puzzle considers if there exists a simple symbolic formula for the following: 1+(1/2)^3+(1/3)^3+(1/4)^3+. . . . But why is this issue―the sum of the reciprocals of the positive integers cubed―so important? With In Pursuit of Zeta-3, popular math writer Paul Nahin investigates the history and significance of this mathematical conundrum.
Drawing on detailed examples, historical anecdotes, and even occasionally poetry, Nahin sheds light on the richness of the nature of zeta-3. He shows its intimate connections to the Riemann hypothesis, another mathematical mystery that has stumped mathematicians for nearly two centuries. He looks at its links with Euler’s achievements and explores the modern research area of Euler sums, where zeta-3 occurs frequently. An exact solution to the zeta-3 question wouldn’t simply satisfy pure mathematical interest: it would have critical ramifications for applications in physics and engineering, such as quantum electrodynamics. Challenge problems with detailed solutions and MATLAB code are included at the end of each of the book’s sections.
Detailing the trials and tribulations of mathematicians who have approached one of the field’s great unsolved riddles, In Pursuit of Zeta-3 will tantalize curious math enthusiasts everywhere.
Author(s): Paul J. Nahin
Edition: 1
Publisher: Princeton University Press
Year: 2021
Language: English
Commentary: Vector PDF
Pages: 344
City: Princeton, NJ
Tags: Popular Science; Mathematics; Calculus
Cover
Contents
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 ζ(3)
2. More Wizard Math and the Zeta Function ζ(s)
2.1. Euler’s Infinite Series for ζ(2)
2.2. The Beta Function and the Duplication Formula
2.3. Euler Almost Computes ζ(3)
2.4. Integral Forms of ζ(2) and ζ(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 Series and Parseval’s Power Formula
3.4. Calculating ζ(2n) with Fourier Series
3.5. How Fourier Series Fail to Compute ζ(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 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
Appendix 3: Proof That Equals Zero
Appendix 4: Double Integration Reversal Isn’t Always Legal
Appendix 5: Impossibility Results from Computer Science
Challenge Problem Solutions
Acknowledgments
Index
