The fractional sum of small arithmetic functions

Abstract

Motivated by recent results, we study sums of the form $S_f(x)={\sum }_{n\le x}f\left(\lfloor \frac{x}{n}\rfloor \right)$, where $f$ is an arithmetic function and $\lfloor \cdot \rfloor$ denotes the greatest integer function. We show how the error term in the asymptotic formula for $S_f(x)$ can be improved in some specific cases.

7 pages