On some sums involving small arithmetic functions

Let $f$ be any arithmetic function and define $S_f(x):={\sum }_{n\le x}f([x\mathrm{/}n])$. If the function $f$ is small, namely, $f(n)\ll n^{\epsilon },$ then the error term $E_f(x)$ in the asymptotic formula of $S_f(x)$ has the form $O(x^{1\mathrm{/}2+\epsilon })\mathrm{.}$ In this paper, we shall study the mean square of $E_f(x)$ and establish some new results of $E_f(x)$ for some special functions.