On the Rankin-Selberg problem in short intervals

Abstract

If $$\Delta (x):=\sum _{n⩽x}{c}_n-Cx(C>0)$$ 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 Lindelöf hypothesis for $\zeta (s)$, it is shown that $$ \int_X^{2X} \Bigl(Δ(x+U)-Δ(x)\Bigr)^2\,{\roman d} x \;\ll_ε\; X^{9/7+ε}U^{8/7}, $$ while under the Lindelöf 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.

12 pages

References in corpus (1)

• On some mean square estimates in the Rankin-Selberg problem