Remarks on a special value of the Selberg zeta function

2009/02/24 by Nicolas Templier, Templier, Nicolas
Mathematics · #11L05 #11M36 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory (math.SP) #math.NT #math.SP #msc:11L05 #msc:11M36

paper · pdf · doi:10.48550/arxiv.0902.4225

12 pages

openalex publication_date 2009/02/24 · arxiv created 2009/02/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \CmZ_Y0(N) be the constant term of the logarithmic derivative at s=1 of the Selberg zeta function of the modular curve Y₀(N). Jorgenson and Kramer established the bound \CmZ_Y0(N)=O_ε(N^ε), ε>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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