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Asymptotic zeros of Poincaré series

Published 12 Jan 2024 in math.NT | (2401.06739v2)

Abstract: We study the zeros of Poincar\'e series Pk,mP_{k,m} for the full modular group. We consider the case where m∼αkm \sim \alpha k for some constant $\alpha > 0$. We show that in this case a positive proportion of the zeros lie on the line 12+it\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 α\alpha is small.

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