The mean value of symmetric square L-functions

: Olga Balkanova, Dmitry Frolenkov

Publisher: MATHEMATICAL SCIENCE PUBL

: 2018

: Algebra and Number Theory

: ALGEBRA & NUMBER THEORY

: ALGEBR NUMBER THEORY

: 12

: 1

: 35

: 59

: 25

: 1937-0652

DOI: https://doi.org/10.2140/ant.2018.12.35

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.