Power Moments of the Riesz Mean Error Term of Symmetric Square L-Function in Short Intervals

Abstract

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2,Z) and let L(s,sym2f)=∑n=1∞cnn−s,ℜs>1 denote the symmetric square L-function of f. In this paper, we consider the Riesz mean of the form Dρ(x;sym2f)=L(0,sym2f)Γ(ρ+1)xρ+Δρ(x;sym2f) and derive the asymptotic formulas for ∫T−HT+HΔρk(x;sym2f)dx, when k≥3.