Sur la fonction Z\^eta associ\'ee au Laplacien singulier $\Delta_{{\mathcal{O}(m)}_\infty}$

Abstract

In a previous paper, we computed the spectrum of the singular Laplacian attached to canonical metrics on $\mathbb{P}^1$. In this article, we study $\zeta_\infty$, the Zeta function associated to this spectrum. We prove that it admits a meromorphic continuation to $\mathbb{C}$. Moreover, this continuation is holomorphic at zero. We compute $\zeta_\infty(0)$ and $\zeta'_\infty(0)$.