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This short note for non-experts means to demystify the tasks of evaluatingthe Riemann Zeta Function at nonnegative 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 theZeta function, and explain how both the functional equation and all `classical'integer values can be obtained in one sweep using only Euler's favorite methodof generating functions. As a counter-point, I also present an even simplerargument essentially due to Bernhard Riemann, which however requires Cauchy'sresidue theorem, a result not yet available to Euler. As a final point, Iendeavor to clarify how these two methods are organically linked and can betaught as an intuitive gateway into the world of Zeta functionology.

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