On the mean value of symmetric square L-functions

TL;DR

This paper refines the asymptotic understanding of the first moment of symmetric-square L-functions at the critical point, revealing an additional main term and improving error bounds under certain hypotheses.

Contribution

It proves the existence of an extra main term in the asymptotic formula and improves error bounds for the twisted first moment of symmetric-square L-functions.

Findings

Identified an additional main term of size $k^{-1/2}$ in the asymptotics.

Improved error bounds to $l^{5/6+ ext{epsilon}}k^{-1/2+ ext{epsilon}}$ unconditionally.

Under Lindelöf hypothesis, error bounds are further improved to $l^{1/2+ ext{epsilon}}k^{-1/2}$.

Abstract

This paper studies 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 2005 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 Ming Ho 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 Lindel\"{o}f hypothesis for quadratic Dirichlet $L$-functions.