Algebra & Number Theory

The mean value of symmetric square $L$-functions

Olga Balkanova and Dmitry Frolenkov

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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 l k 1 2 + ϵ . We improve the error bound to l 5 6 + ϵ k 1 2 + ϵ unconditionally and to l 1 2 + ϵ k 1 2 under the Lindelöf hypothesis for quadratic Dirichlet L -functions.

Article information

Algebra Number Theory, Volume 12, Number 1 (2018), 35-59.

Received: 20 October 2016
Revised: 30 June 2017
Accepted: 15 November 2017
First available in Project Euclid: 4 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F12: Automorphic forms, one variable
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 34E05: Asymptotic expansions 34E20: Singular perturbations, turning point theory, WKB methods

symmetric square $L$-functions weight aspect Gauss hypergeometric function Liouville–Green method WKB approximation


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.

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