Asymptotic zeros of Poincaré series

2024/01/12 by Noam Kimmel, Kimmel, Noam · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2401.06739

openalex publication_date 2024/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the zeros of Poincaré series P_k,m for the full modular group. We consider the case where m ∼ αk for some constant α> 0. We show that in this case a positive proportion of the zeros lie on the line (1)/(2) + it. We further show that if α> (log(2))/(2π) then the imaginary axis also contains a positive proportion of zeros. We also give a description for the location of the non-real zeros when α is small.

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