Non-vanishing of Maass form 𝐿-functions at the central point

Balkanova, Olga; Huang, Bingrong; Södergren, Anders

doi:10.1090/proc/15208

Non-vanishing of Maass form $L$-functions at the central point
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by Olga Balkanova, Bingrong Huang and Anders Södergren PDF

Proc. Amer. Math. Soc. 149 (2021), 509-523 Request permission

Abstract:

In this paper, we consider the family $\{L_j(s)\}_{j=1}^{\infty }$ of $L$-functions associated to an orthonormal basis $\{u_j\}_{j=1}^{\infty }$ of even Hecke–Maass forms for the modular group $\operatorname {SL}(2,\mathbb Z)$ with eigenvalues $\{\lambda _j=\kappa _{j}^{2}+1/4\}_{j=1}^{\infty }$. We prove the following effective non-vanishing result: At least $50 \%$ of the central values $L_j(1/2)$ with $\kappa _j \leq T$ do not vanish as $T\rightarrow \infty$. Furthermore, we establish effective non-vanishing results in short intervals.

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