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

Olga Balkanova; Bingrong Huang; Anders Södergren

doi:10.1090/proc/15208

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

Olga Balkanova, Bingrong Huang, Anders Södergren

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we consider the family {L_j(s)}^∞_j₌₁ of L-functions associated to an orthonormal basis {u_j}^∞_j₌₁ of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λ_j = κ²_j + 1/4}^∞_j=1. We prove the following effective non-vanishing result: At least 50% of the central values L_j(1/2) with κ_j ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals.

Original language	English
Pages (from-to)	509-523
Number of pages	15
Journal	Proceedings of the American Mathematical Society
Volume	149
Issue number	2
DOIs	https://doi.org/10.1090/proc/15208
State	Published - Feb 2021
Externally published	Yes

Keywords

L-functions
Maass cusp forms
Mollification
Non-vanishing

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10.1090/proc/15208

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title = "Non-vanishing of maass form L-functions at the central point",

abstract = "In this paper, we consider the family {Lj(s)}∞j=1 of L-functions associated to an orthonormal basis {uj}∞j=1 of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λj = κ2j + 1/4}∞j=1. We prove the following effective non-vanishing result: At least 50% of the central values Lj(1/2) with κj ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals.",

keywords = "L-functions, Maass cusp forms, Mollification, Non-vanishing",

author = "Olga Balkanova and Bingrong Huang and Anders S{\"o}dergren",

note = "Publisher Copyright: {\textcopyright} 2020 American Mathematical Society",

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AU - Balkanova, Olga

AU - Huang, Bingrong

AU - Södergren, Anders

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N2 - In this paper, we consider the family {Lj(s)}∞j=1 of L-functions associated to an orthonormal basis {uj}∞j=1 of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λj = κ2j + 1/4}∞j=1. We prove the following effective non-vanishing result: At least 50% of the central values Lj(1/2) with κj ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals.

AB - In this paper, we consider the family {Lj(s)}∞j=1 of L-functions associated to an orthonormal basis {uj}∞j=1 of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λj = κ2j + 1/4}∞j=1. We prove the following effective non-vanishing result: At least 50% of the central values Lj(1/2) with κj ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals.

KW - L-functions

KW - Maass cusp forms

KW - Mollification

KW - Non-vanishing

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

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