Remarks on a special value of the Selberg zeta function

Details Remarks on a special value of the Selberg zeta function Remarks on a special value of the Selberg zeta function

2009-02-24

arxiv.org/abs/0902.4225v2

Mathematics

Number Theory

12 pages

Scientific paper

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.

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