On Some Sums Involving Small Arithmetic Functions

Let f be any arithmetic function and define Sf(x):=& sum;n <= xf([x/n])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_{f}}(x):=\sum\nolimits_{{n \le x}}f([x/n])$$\end{document}. If the function f is small, namely, f(n) << n epsilon, then the error term Ef(x) in the asymptotic formula of Sf(x) has the form O(x1/2+epsilon). In this paper, we shall study the mean square of Ef(x) and establish some new results of Ef(x) for some special functions.