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

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].