A1 Refereed original research article in a scientific journal
The mean value of symmetric square L-functions
Authors: Olga Balkanova, Dmitry Frolenkov
Publisher: MATHEMATICAL SCIENCE PUBL
Publication year: 2018
Journal: Algebra and Number Theory
Journal name in source: ALGEBRA & NUMBER THEORY
Journal acronym: ALGEBR NUMBER THEORY
Volume: 12
Issue: 1
First page : 35
Last page: 59
Number of pages: 25
ISSN: 1937-0652
DOI: https://doi.org/10.2140/ant.2018.12.35
Abstract
We study the first moment of symmetric-square L-functions at the critical point in the weight aspect. Asymptotics with the best known error term O(k(-1/2)) were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size k(-1/2) in the asymptotic formula and show that the remainder term decays exponentially in k. The twisted first moment was evaluated asymptotically by Ng with the error bounded by lk(1/2+epsilon). We improve the error bound to l(5/6+epsilon)k(-1/2+epsilon) unconditionally and to l(-1/2+epsilon)k(-1/2) under the Lindelof hypothesis for quadratic Dirichlet L-functions.
We study the first moment of symmetric-square L-functions at the critical point in the weight aspect. Asymptotics with the best known error term O(k(-1/2)) were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size k(-1/2) in the asymptotic formula and show that the remainder term decays exponentially in k. The twisted first moment was evaluated asymptotically by Ng with the error bounded by lk(1/2+epsilon). We improve the error bound to l(5/6+epsilon)k(-1/2+epsilon) unconditionally and to l(-1/2+epsilon)k(-1/2) under the Lindelof hypothesis for quadratic Dirichlet L-functions.
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