Classical values of Zeta, as simple as possible but not simpler

TL;DR

This paper simplifies the understanding of the classical values of the Riemann Zeta Function at specific points, using Euler's generating functions and Riemann's residue theorem to connect historical methods.

Contribution

It presents a unified, accessible explanation of evaluating the Zeta function at non-positive integers and even natural numbers, linking Euler's and Riemann's approaches.

Findings

Euler's generating functions derive classical Zeta values.

Riemann's residue theorem offers a simpler alternative.

Both methods are interconnected and pedagogically valuable.

Abstract

This short note for non-experts means to demystify the tasks of evaluating the Riemann Zeta Function at non-positive integers and at even natural numbers, both initially performed by Leonhard Euler. Treading in the footsteps of G. H. Hardy and others, I re-examine Euler's work on the functional equation for the Zeta function, and explain how both the functional equation and all `classical' integer values can be obtained in one sweep using only Euler's favorite method of generating functions. As a counter-point, I also present an even simpler argument essentially due to Bernhard Riemann, which however requires Cauchy's residue theorem, a result not yet available to Euler. As a final point, I endeavor to clarify how these two methods are organically linked and can be taught as an intuitive gateway into the world of Zeta functionology.