Certain floor function sums over powers

TL;DR

This paper investigates sums involving floor functions over powers, providing new estimates for error terms when the function is supported on rth powers, and discusses related conjectures.

Contribution

It introduces novel bounds for error terms in sums over powers and formulates conjectures related to these estimates.

Findings

Nontrivial estimates for error terms in sums over powers

Results applicable to functions supported on rth powers with r≥2

Several conjectures proposed for future research

Abstract

We study the sums $$ S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor\right) $$ when $f$ is supported on $r$th powers with $r\geq 2$. This restriction allows us to give nontrivial estimates for one of the error terms in the asymptotic expansion of $S_f(x)$. We also state several conjectures related to our results.