Asymptotic zeros of Poincar\'e series

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.