The fractional sum of small arithmetic functions

TL;DR

This paper investigates sums involving arithmetic functions evaluated at the integer part of x/n, providing improved error bounds in their asymptotic formulas for certain cases.

Contribution

It introduces new techniques to refine the error estimates in the asymptotic analysis of fractional sums of small arithmetic functions.

Findings

Improved error bounds for specific arithmetic functions

Asymptotic formulas for sums involving floor functions

Enhanced understanding of fractional sums in number theory

Abstract

Motivated by recent results, we study sums of the form $S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor \right)$, where $f$ is an arithmetic function and $\left\lfloor\cdot\right\rfloor$ 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.