Revisiting the Riemann Zeta Function at Positive Even Integers

When/Where:

April 5, 2016, 3:00 — 3:50pm at LIT 368.

Abstract:

In his number theory seminar course, Professor Alladi provided a simple inductive proof that \(\zeta(2k)\) is a rational multiple of \(\pi^{2k}\) for each positive integer \(k\). The argument relies on little more than Parseval’s Identity and basic calculus. In this talk, we will begin by proving a new identity involving Bernoulli numbers. We then describe how to use this identity in order to extend Professor Alladi’s argument and obtain a new proof of Euler’s explicit formula for \(\zeta(2k)\).