On the fourth moment in the Rankin-Selberg problem

Abstract

If $$\Delta (x):=\sum _{n\le x}{c}_n-Cx$$ 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 ${\Delta }_1(x)={\int }_0^x\Delta (u)du$. The latter bound is, up to `$\epsilon$', best possible.

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