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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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:
  • 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:
  • 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:
  • 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
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 509-523
  • MSC (2020): Primary 11F67, 11F12
  • DOI:
  • MathSciNet review: 4198061