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

In this paper, we consider sums related to the floor function. We improve previous results for some special arithmetic functions considered by O. Bordelles [On certain sums of number theory, Int. J. Number Theory, 18(9):2053- 2074, 2022], J. Stucky [The fractional sum of small arithmetic functions, J. Number Theory, 238:731-739, 2022], and J. Wu [Note on a paper by Bordelles, Dai, Heyman, Pan and Shparlinski, Period. Math. Hung., 80(1):95-102, 2020]. It is worth emphasizing that we use much simpler methods to give much better results than previous.