J. Korean Math. Soc. 2023; 60(1): 167-193
Online first article December 15, 2022 Printed January 1, 2023
https://doi.org/10.4134/JKMS.j220242
Copyright © The Korean Mathematical Society.
Peter Jaehyun Cho , Gyeongwon Oh
Ulsan National Institute of Science and Technology; Ulsan National Institute of Science and Technology
Let $f$ be a self-dual primitive Maass or modular forms for level $4$. For such a form $f$, we define \begin{align*} N_f^s(T)\!:=\!|\{\rho \in \mathbb{C} : |\Im(\rho)| \leq T, \text{ $\rho$ is a non-trivial simple zero of $L_f(s)$} \}|. \end{align*} We establish an omega result for $N_f^s(T)$, which is $N_f^s(T)=\Omega \big( T^{\frac{1}{6}-\epsilon} \big)$ for any $\epsilon>0$. For this purpose, we need to establish the Weyl-type subconvexity for $L$-functions attached to primitive Maass forms by following a recent work of Aggarwal, Holowinsky, Lin, and Qi.
Keywords: Simple zero, Maass forms, Weyl-type subconvexity
MSC numbers: Primary 11M26, 11L07
Supported by: Peter J. Cho is supported by the NRF grant funded by the Korea government(MSIT) (No. 2019R1F1A1062599) and the Basic Science Research Program(2020R1A4A1016649).
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