Classical Values of Zeta, as Simple as Possible But Not Simpler

This is the authors' abstract. We don't add key points for this paper.

This short note for non-experts means to demystify the tasks of evaluating the Riemann Zeta Function at nonpositive 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. History in a nutshellThe notorious Basel problem, posed by Pietro Mengoli in 1650, asked for the precise evaluation of the infinite sumwhich a modern reader would instantly recognize as the value of the celebrated Riemann Zeta function at the point 2.This problem was solved by Leonhard Euler by 1734, with his solution read at The Saint Petersburg Academy of Sciences on December 5, 1734.Euler's investigation of this and related problems began in 1731 and continued at least till 1749.Euler's tour de force led to his solution or, in fact, multiple solutions to the Basel problem, followed by his successful evaluation of ζ at all positive even integers and at all negative integers, his infinite product representation for ζ, and his lesser-known formulation of its functional equation, albeit only with a partial proof.The fascinating history of Euler's work on this subject is laid out in [2].About a century later in his groundbreaking memoir [16] (for an English translation, see [5, pp.190-198]) Bernhard Riemann re-introduced ζ as a function of a complex variable s, established a number of its integral representations, found and justified its analytic continuation beyond the half-plane Re s > 1, proved its functional equation, and formulated The Riemann Hypothesis.Riemann's memoir must have been the most influential nine pages of mathematics ever written.