On the Rankin-Selberg problem in short intervals

TL;DR

This paper establishes new upper bounds for the mean square of the error term in the Rankin-Selberg problem over short intervals, under assumptions like the Lindelöf hypothesis, advancing understanding of its fluctuations.

Contribution

It provides non-trivial bounds for the mean square of the error term in the Rankin-Selberg problem in short intervals, under hypotheses such as Lindelöf, which was previously unexplored.

Findings

Under Lindelöf hypothesis, mean square bound is $X^{9/7+ ext{epsilon}}U^{8/7}$.

Under Rankin-Selberg Lindelöf hypothesis, the bound improves to $X^{1+ ext{epsilon}}U^{4/3}$.

Analogous bounds are established for the discrete second moment.

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.