Zeta functions: from classical generating functions to abstract harmonic analysis

Date: Tuesday 4 October 2022

Abstract

The modern practitioner of zeta functions tends to work with Haar integrals over locally compact groups. This is a far cry from our familiar representation of the Riemann zeta function as a Dirichlet series or as an Euler product. Nonetheless, Tate’s Thesis (1950) makes a convincing case for adopting this more abstract perspective. It turns out that the study of many other classical generating functions can be enriched through abstract harmonic analysis e.g. Cauchy’s q-binomial series, Eisenstein-Hermite’s integral lattice generating function, Euler’s partition generating function, and Erdos-Szekeres’ abelian group generating function. We see this by relating our classical generating functions to zeta functions of arithmetic modules, which are originally defined as Dirichlet series.