Elegant Techniques for Zeta and Harmonic Series: A new accessible approach to the Basel problem, Euler sums, and Plouffe identities (Paperback) (ISBN-13: 9788409841486)

Rediscover Zeta and Harmonic Series Through a Bold New Algebraic Perspective

The Basel Problem was famously solved by Euler, who unveiled the beauty of even zeta values and odd beta values. Since then, the landscape of infinite series has expanded through the work of many great mathematicians like Dirichlet, Ramanujan, or Zagier, among many others-often requiring advanced and heavy machinery.

What if there was a simpler way?

This book introduces a compelling alternative technique that naturally unifies these famous results. By focusing on the algebraic interrelations between quantities, this method sheds new light on the links between particular integer zeta values, Euler sums, and Plouffe identities.

The best part? You'll find yourself absorbed while exploring the different paths each chapter opens.

Whether you are a student or a seasoned researcher, this book provides a fresh toolkit for evaluating series and leaves you with open questions ready to be taken to the next frontier of mathematical discovery.

• Introduction: we cover the motivation for the problem of evaluating the zeta function at 2, 3, 4, 5, ...
• Ch2 - Solving the Basel problem: we introduce the new technique, develop the relations between all zeta(2n), and then solve the Basel challenge.
• Ch3 - Basic harmonic series: we find all the simple Euler sums of harmonic numbers and the skew harmonic version, over all powers of natural and odd numbers. We also cover alternating sums and series with odd-based harmonics, with partial results.
• Ch4 - Higher order harmonic series: we extend the results to series that contain higher order harmonics over natural and odd denominators. We cover how the technique covers some powers and products, and a few generalizations.
• Ch5 - Quadratics and pure zeta-family identities: we go back to the initial quadratic identity developed at chapter 2, do a deeper exploration and expand and find other quadratic identities. We cover the odd-integer 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 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 and 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 special limit: we look into more detail to a limit that acts as general example 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.