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

TL;DR

This paper investigates the spectral properties of a singular Laplacian on the projective line, focusing on the associated Zeta function, proving its meromorphic continuation, and computing key special values.

Contribution

It extends previous work by analyzing the Zeta function linked to the singular Laplacian, providing its meromorphic continuation and explicit evaluations at zero.

Findings

Zeta function admits meromorphic continuation to a7.

The continuation is holomorphic at zero.

Explicit formulas for a7_e9(0) and a7'_e9(0) are obtained.

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 $\zeta'_\infty(0)$.