On general sums involving the floor function with applications to $k$-free numbers

TL;DR

This paper improves and refines results on sums involving the floor function, with applications to $k$-free numbers, advancing understanding of related arithmetic functions and their bounds.

Contribution

It provides improved bounds and refined results for sums involving the floor function, extending previous work on special and general arithmetic functions.

Findings

Enhanced bounds for sums involving the floor function.

Refined results for arithmetic functions related to $k$-free numbers.

Generalization of previous results with improved accuracy.

Abstract

In this paper, we consider sums related the floor function. We can improve some previous results for some special arithmetic functions considered by Bordell\'es [3], Stucky[9] and Liu-Wu-Yang [6]. We can also give a refined result for previous results for general function considered by Bordell\'es, Dai, Heyman, Pan and Shparlinski [1] and Wu [11].