Asymptotic zeros of Poincar\'e series

Noam Kimmel

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arXiv · 2401.06739

Asymptotic zeros of Poincar\'e series

Noam Kimmel

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Abstract

We study the zeros of Poincar\'e series $P_{k,m}$ for the full modular group. We consider the case where $m \sim \alpha k$ for some constant $\alpha > 0$. We show that in this case a positive proportion of the zeros lie on the line $\frac{1}{2} + it$. We further show that if $\alpha > \frac{\log(2)}{2\pi}$ 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 $\alpha$ is small.

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Noam Kimmel. 2024-01-12. Asymptotic zeros of Poincar\'e series. https://arxiv.org/abs/2401.06739

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