On Dirichlet eigenvalues of regular polygons

TL;DR

This paper derives an asymptotic expansion for the first Dirichlet eigenvalue of regular polygons as the number of sides increases, linking the coefficients to multiple zeta values and explicitly computing them up to n=14.

Contribution

It provides a novel asymptotic expansion for Dirichlet eigenvalues of regular polygons and explicitly computes the expansion coefficients involving multiple zeta values.

Findings

Asymptotic expansion of eigenvalues as N→∞

Explicit polynomial coefficients up to n=14

Connection to multiple zeta values

Abstract

We prove that the first Dirichlet eigenvalue of a regular $N$-gon of area $\pi$ has an asymptotic expansion of the form $\lambda_1(1+\sum_{n\ge3}C_n(\lambda_1)N^{-n})$ 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 $n\le14$.