On Dirichlet eigenvalues of regular polygons

Abstract

We prove that the first Dirichlet eigenvalue of a regular N -gon of area π has an asymptotic expansion of the form λ 1 (1 + n≥3 Cn(λ1) N n ) as N → ∞, where λ 1 is the first Dirichlet eigenvalue of the unit disk and C n are polynomials whose coefficients belong to the space of multiple zeta values of weight n. We also explicitly compute these polynomials for all n ≤ 14.