The third-power moment of the Riesz mean error term of symmetric square L-function

Abstract

Let $ f(z) $ be a holomorphic Hecke eigenform of weight $ k $ with respect to the full modular group $ SL(2, Z) $, and let $ \Delta_{\rho}(x; sym^{2}f) $ be the error term of the Riesz mean of the symmetric square $ L $-function $ L(s, sym^{2}f) $. In this paper, using a Voronoï type formula for $ \Delta_{\rho}(x; sym^{2}f) $, we consider the third-power moment of $ \Delta_{\rho}(x; sym^{2}f) $ and derive the asymptotic formula for $ \int_{1}^{T}\Delta_{\rho}^{3}(x;sym^{2}f)dx. $

Keywords

• ymmetric square l-function
• power moment
• riesz mean
• voronöi type formula