Elegant Techniques for Zeta and Harmonic Series
A new accessible approach to the Basel problem, Euler sums and Plouffe Identities
Unifying the Basel problem, Harmonic sums, and more
Same technique, different results.
Learn how a new technique unifies the solution to problems like the exact evaluation of \(\zeta(2n),\, n\in\mathbb{N}\), \(\beta(2n-1),\, n\in\mathbb{N}\), harmonic sums (Euler sums), Plouffe identities and more.
No contour integration, no integrals, no trigonometric or special functions, just algebra and basic calculus to begin the journey — and yes, Plouffe identities follow from Euler's sum of squares, well advanced in the topic.
But the best of all is that more problems are uncovered than solved. On every chapter, every curious reader will find themselves further exploring the topic beyond what's in the book.
What's Inside this book
We will cover different topics related with certain types of infinite series, specially from the algebraic point of view and with the help of some calculus (a few limits and straightforward derivatives).
These are the chapters and the topics covered:
- Introduction: we cover the motivation for the problem with evaluation of the zeta function at natural number arguments.
- Ch2 — Solving the Basel problem: we introduce the technique, develop the relations between all \(\zeta(2n),\, n\in\mathbb{N}\), and then solve the Basel challenge.
- Ch3 — Basic harmonic series: we find all the simple Euler sums of harmonic numbers \(H_n\) and the skew harmonic version, over all powers of natural and odd numbers. We also cover alternating sums and series with the odd-based harmonic \(O_n\) with partial results.
- Ch4 — Higher order harmonic series: we extend the results to series that contain instances of \(H_n^{(k)}\) and their skew version, over natural and odd denominators. We cover how the technique covers some powers like \(H_n^2\) and products, and a few generalizations.
- Ch5 — Quadratics and pure zeta-family identities: we go back to the initial identity developed at the beginning of the book, do a deeper exploration and expand and find other quadratic identities. We cover the odd-parameter betas, and end up considering all products between the zeta family values and do some other explorations.
- Ch6 — Fourth degree series: the shape of the functions suggest this next exploration, where we find, thanks to Euler, the connection with hyperbolic cotangent. There's room to still get alternative pure-zeta identities, others containing harmonics and more.
- Ch7 — Plouffe-Ramanujan identities: using the results from previous chapter we prove many of Plouffe's identities and generalize some related Ramanujan identities, covering not only odd-argument zetas but also even-argument betas.
- Ch8 — Digamma explorations: in this chapter we unify previous results and find further examples of these zeta series representations. We also find quadratic closed forms free of harmonics or other artifacts.
- Ch9 — Higher degree series: we retake exploration of cubics, sixties and 8th degree identities.
- Ch10 — Exploring series with factorials: we generalize some key concepts from Chapter 2 and apply to series based on factorials, developing new and surprising identities. In this chapter we use integration techniques a few times.
- Ch11 — Inspiring infinite product: following Euler's ideas, we develop generic series identities that allow us to obtain general higher-order harmonic sums, complementing previous chapters' results.
- Appendix A — A couple of limits: we look into more detail to a couple of limits that act as general examples needed to obtain some results.
- Appendix B — Sums tables: summary of a certain type of sums tabulated for low degrees, along with methods to obtain them.
While we will cover a well understood topic, we will also consider other less-known identities and sums. Take, for example, the next as a typical identity from the book:
Or the next one, a bit more complex, Plouffe-style identity (where \(N=2n-1\), \(a\in\mathbb{R}\)):
About me
I graduated in mathematics in the early 2000's, but never worked professionally in professional mathematics. Instead, I've dedicated my career to software development, disconnected from the academic world. Despite that, I've been always fascinated by infinite sums and especially intrigued by numbers like the Apéry constant. In the last four years, I've spent the first hour of every morning working on this problem and writing the present book.