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

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.