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

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- by Olga Balkanova, Bingrong Huang and Anders Södergren PDF
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**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.## References

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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
- 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: https://doi.org/10.1090/proc/15208
- MathSciNet review: 4198061