On the mean value of symmetric square L-functions

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 $l{k}^{-1/2+\epsilon }$. 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 Lindelöf hypothesis for quadratic Dirichlet $L$-functions.

30 pages; Remark on page 4 and new reference added

Cited by in corpus (5)

• Moments and hybrid subconvexity for symmetric-square L-functions
• Bounds for a spectral exponential sum
• A Mean Value Result for a Product of GL(2) and GL(3) L-Functions
• Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions
• The second moment of symmetric square L-functions over Gaussian integers