On Some Mean Value Results for the Zeta-Function and a Rankin–Selberg Problem

Denote by Δ1(x;φ) the error term in the classical Rankin–Selberg problem. Denote by ζ(s) the Riemann zeta-function. We establish an upper bound for this integral ∫0TΔ1(t;φ)ζ12+it2dt. In addition, when 2≤k≤4 is a fixed integer, we will derive an asymptotic formula for the integral ∫1TΔ1k(t;φ)ζ12+it2dt. The results rely on the power moments of Δ1(t;φ) and E(t), the classical error term in the asymptotic formula for the mean square of ζ12+it.