The mean value of symmetric square L-functions

Balkanova, Olga; Frolenkov, Dmitry

doi:10.2140/ant.2018.12.35

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The mean value of symmetric square $L$-functions

Olga Balkanova and Dmitry Frolenkov

Vol. 12 (2018), No. 1, 35–59

DOI: 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 $l k^{- 1 ∕ 2 + ϵ}$ . 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 11F12

Secondary: 33C05, 34E05, 34E20

Milestones

Received: 20 October 2016

Revised: 30 June 2017

Accepted: 15 November 2017

Published: 13 March 2018

Authors

Olga Balkanova
Department of Mathematics and Statistics University of Turku Turku Finland

Dmitry Frolenkov
National Research University Higher School of Economics Steklov Mathematical Institute of Russian Academy of Sciences Moscow Russia

Vol. 12, No. 1, 2018