On Some Sums Involving Small Arithmetic Functions

Let f be any arithmetic function and define \({S_{f}}(x):=\sum\nolimits_{{n \le x}}f([x/n])\) $$S_f(x):={\sum }_{n\le x}f([x/n])$$ . If the function f is small, namely, f(n) ≪ n^ε, then the error term E_f(x) in the asymptotic formula of S_f(x) has the form O(x^1/2+ε). 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.