On the Rankin-Selberg problem in short intervals

Abstract

If $$ \Delta(x) \;:=\; \sum_{n\leqslant x}c_n - Cx\qquad(C>0) $$ denotes the error term in the classical Rankin-Selberg problem, then we obtain a non-trivial upper bound for the mean square of $\Delta(x+U) - \Delta(x)$ for a certain range of $U = U(X)$. In particular, under the Lindel\"of hypothesis for $\zeta(s)$, it is shown that $$ \int_X^{2X} \Bigl(\Delta(x+U)-\Delta(x)\Bigr)^2\,{\roman d} x \;\ll_\epsilon\; X^{9/7+\epsilon}U^{8/7}, $$ while under the Lindel\"of hypothesis for the Rankin-Selberg zeta-function the integral is bounded by $X^{1+\epsilon}U^{4/3}$. An analogous result for the discrete second moment of $\Delta(x+U)-\Delta(x)$ also holds.