On a sum involving general arithmetic functions and the integral part function

Abstract

Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity \[ S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where $[t]$ is the integral part of the real number $t$. This generalizes some recent results of Bordell\`es, Dai, Heyman, Pan and Shparlinski.