Vol. 12, No. 1, 2018

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The mean value of symmetric square $L$-functions

Olga Balkanova and Dmitry Frolenkov

Vol. 12 (2018), No. 1, 35–59

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(k12) were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size k12 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 lk12+ϵ. We improve the error bound to l56+ϵk12+ϵ unconditionally and to l12+ϵk12 under the Lindelöf hypothesis for quadratic Dirichlet L-functions.

symmetric square $L$-functions, weight aspect, Gauss hypergeometric function, Liouville–Green method, WKB approximation
Mathematical Subject Classification 2010
Primary: 11F12
Secondary: 33C05, 34E05, 34E20
Received: 20 October 2016
Revised: 30 June 2017
Accepted: 15 November 2017
Published: 13 March 2018
Olga Balkanova
Department of Mathematics and Statistics
University of Turku
Dmitry Frolenkov
National Research University Higher School of Economics
Steklov Mathematical Institute of Russian Academy of Sciences