On the fourth moment in the Rankin-Selberg problem

Abstract

If $$ Δ(x) := \sum_{n\le x}c_n - Cx $$ denotes the error term in the classical Rankin-Selberg problem, then it is proved that $$ \int_0^X Δ^4(x)\d x \ll_εX^{3+ε},\quad \int_0^X Δ_1^4(x)\d x \ll_εX^{11/2+ε}, $$ where $Δ_1(x) = \int_0^xΔ(u) du$. The latter bound is, up to `$ε$', best possible.