Higher power moments of the Riesz mean error term of symmetric square L-function

Abstract

Let Δρ(x;sym2f) be the error term of the Riesz mean of the symmetric square L-function. We give the higher power moments of Δρ(x;sym2f) and show that if there exists a real number A0:=A0(ρ)>3 such that ∫1T|Δρ(x;sym2f)|A0dx≪T1+2ρ+1/3A0+ε, then we can derive asymptotic formulas for ∫1TΔρh(x;sym2f)dx, 3≤h<A0, h∈N{double-struck}. Particularly, we get asymptotic formulas for ∫1TΔ1/2h(x;sym2f)dx, h=3,4,5 unconditionally. © 2011 Elsevier Inc.