On Dirichlet eigenvalues of regular polygons (Q6558339)

Let \(\mathcal{P}_N\) be a regular polygon of area \(\pi\) with \( N \geq 3\) sides. In this paper under review, the authors study the behavior of the Dirichlet eigenvalues \(\lambda_k(\mathcal{P}_N)\) as \(N\) goes to infinity. More precisely, they show that there exists a sequence of polynomials \(C_n\in \mathfrak{Z}_n[\lambda]\), \(n\geq 1\) where \(\mathfrak{Z}_n\) is the space of multiple zeta values of weight \(n\), such that \[\frac{\lambda_k(\mathcal{P}_N)}{\lambda_k}\sim 1+\sum_{n=1}^\infty \frac{C_n(\lambda_k)}{N^n} \]whenever \(\lambda_k\) is a radially-symmetric Dirichlet eigenvalue of the unit disk.\N\NThe approach to prove the above result yields an explicit symbolic algorithm for computing the polynomials \(C_n\). For instance, the authors calculate \(C_n\) for \(n\leq 14\) to check their conjecture which asserts that the polynomial \(C_n(\lambda)\) belongs to the space of singlevalued multiple zeta values of weight \(n\).