Remarks on a special value of the Selberg zeta function

TL;DR

This paper investigates the constant term of the logarithmic derivative of the Selberg zeta function for modular curves, providing new bounds using $L$-functions and exponential sums, especially for prime levels.

Contribution

It offers a novel proof for bounds on the Selberg zeta function's constant term for prime levels, improving previous methods and deriving a logarithmic bound.

Findings

Established an $O(( ext{log } N)^k)$ bound for the constant term for prime $N$

Provided an alternative proof using $L$-functions and exponential sums

Improved upon previous geometric invariant-based bounds

Abstract

Let $\CmZ_{Y_0(N)}$ be the constant term of the logarithmic derivative at $s=1$ of the Selberg zeta function of the modular curve $Y_0(N)$. Jorgenson and Kramer established the bound $\CmZ_{Y_0(N)}=O_\epsilon(N^\epsilon)$, $\epsilon>0$ by relating it to geometric invariants. In this article we give, for $N$ prime, another proof via $L$-functions and exponential sums improving on a previous approach by Abbes-Ullmo and Michel-Ullmo. We further derive a power of $\log N$ bound along the same line.