On some mean square estimates in the Rankin-Selberg problem

An overview of the classical Rankin-Selberg problem involving the asymptotic formula for sums of coefficients of holomorphic cusp forms is given. We also study the function $\Delta(x;\xi) (0\le\xi\le1)$, the error term in the Rankin-Selberg problem weighted by $\xi$-th power of the logarithm. Mean square estimates for $\Delta(x;\xi)$ are proved.