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


Authors: Olga Balkanova, Bingrong Huang and Anders Södergren
Journal: Proc. Amer. Math. Soc. 149 (2021), 509-523
MSC (2020): Primary 11F67, 11F12
DOI: https://doi.org/10.1090/proc/15208
Published electronically: December 7, 2020
MathSciNet review: 4198061
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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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Additional Information

Olga Balkanova
Affiliation: Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina st., Moscow 119991, Russia and Institute of Applied Mathematics, Khabarovsk Division, 54 Dzerzhinsky Street, Khabarovsk 680000, Russia
MR Author ID: 1168196
ORCID: 0000-0003-3427-0300
Email: olgabalkanova@gmail.com

Bingrong Huang
Affiliation: Data Science Institute and School of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
ORCID: 0000-0002-8987-0015
Email: brhuang@sdu.edu.cn

Anders Södergren
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
MR Author ID: 931224
ORCID: 0000-0001-6061-0319
Email: andesod@chalmers.se

Keywords: Maass cusp forms, L-functions, non-vanishing, mollification.
Received by editor(s): October 18, 2018
Received by editor(s) in revised form: December 12, 2018, and May 1, 2020
Published electronically: December 7, 2020
Additional Notes: The first author was supported by the Russian Science Foundation under grant [19-11-00065].
The second author was supported by the European Research Council, under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 786758.
The third author was supported by a grant from the Swedish Research Council (grant 2016-03759).
Communicated by: Amanda Folsom
Article copyright: © Copyright 2020 American Mathematical Society