Dirichlet eigenvalues of regular polygons

Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)

For N>4 the first Dirichlet eigenvalue of a regular N-gon does not seem to have any closed form expression in terms of other known mathematical constants. However, it has been previously observed that it has an asymptotic expansion in powers of 1/N whose low order coefficients can be expressed in terms of special values of the Riemann zeta function at positive integers. It turns out that all coefficients of this expansion can be computed in closed form, but in general they involve more general multiple zeta values that (conjecturally) cannot be expressed in terms of the usual zeta values. I will discuss the proof of this result as well as some curious results and identities that arise in the process. The talk is based on a joint work with David Berghaus, Bogdan Georgiev, and Hartmut Monien.