Revisiting the Riemann Zeta Function at the Positive Even Integers

When/Where:

February 20, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

Euler showed that the values of the Riemann zeta function at positive even integer arguments \(2k\) are rational multiples of \(\pi^{2k}\), these rationals being given in terms of Bernoulli numbers. Over the years, several proofs of this celebrated result of Euler have been given. We will discuss a new proof by simply starting with the determination of the Fourier coefficients of \(f(x)=x^k\), and using the Parseval identity. This leads to a pair of intertwining recurrences, which when investigated closely leads to a very different proof of Euler’s formula and a surprising new identity for Bernoulli numbers.

This is joint work with Colin Defant. Time permitting,

I will also discuss connections between Bernoulli polynomials and the analytic continuation of the Riemann zeta function.