Mathematics > Number Theory
[Submitted on 18 Oct 2018 (v1), last revised 12 Dec 2018 (this version, v2)]
Title:Non-vanishing of Maass form L-functions at the critical point
View PDFAbstract: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 $SL(2, Z)$ 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.
Submission history
From: Olga Balkanova [view email][v1] Thu, 18 Oct 2018 11:23:35 UTC (12 KB)
[v2] Wed, 12 Dec 2018 11:02:55 UTC (13 KB)
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