On Dirichlet eigenvalues of regular polygons

Abstract

We prove that the first Dirichlet eigenvalue of a regular $N$-gon of area $\pi$ has an asymptotic expansion of the form ${\lambda }_1(1+{\sum }_{n\ge 3}{C}_n({\lambda }_1)N^{-n})$ as $N\to \infty$, where ${\lambda }_1$ 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 $n\le 14$.

15 pages