Algebra & Number Theory
- Algebra Number Theory
- Volume 12, Number 1 (2018), 35-59.
The mean value of symmetric square $L$-functions
Olga Balkanova and Dmitry Frolenkov
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Abstract
We study the first moment of symmetric-square -functions at the critical point in the weight aspect. Asymptotics with the best known error term were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size in the asymptotic formula and show that the remainder term decays exponentially in . The twisted first moment was evaluated asymptotically by Ng with the error bounded by . We improve the error bound to unconditionally and to under the Lindelöf hypothesis for quadratic Dirichlet -functions.
Article information
Source
Algebra Number Theory, Volume 12, Number 1 (2018), 35-59.
Dates
Received: 20 October 2016
Revised: 30 June 2017
Accepted: 15 November 2017
First available in Project Euclid: 4 April 2018
Permanent link to this document
https://projecteuclid.org/euclid.ant/1522807230
Digital Object Identifier
doi:10.2140/ant.2018.12.35
Mathematical Reviews number (MathSciNet)
MR3781432
Zentralblatt MATH identifier
06861735
Subjects
Primary: 11F12: Automorphic forms, one variable
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 34E05: Asymptotic expansions 34E20: Singular perturbations, turning point theory, WKB methods
Keywords
symmetric square $L$-functions weight aspect Gauss hypergeometric function Liouville–Green method WKB approximation
Citation
Balkanova, Olga; Frolenkov, Dmitry. The mean value of symmetric square $L$-functions. Algebra Number Theory 12 (2018), no. 1, 35--59. doi:10.2140/ant.2018.12.35. https://projecteuclid.org/euclid.ant/1522807230
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