On fractional sums of the divisor functions

Abstract

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by [O. Bordellés, On certain sums of number theory, Int. J. Number Theory 18(9) (2022) 2053–2074; K. Liu, J. Wu and Z. S. Yang, On some sums involving the integral part function, preprint (2021); arXiv:2109.01382v1 [math.NT]]. Precisely, we can show that [Formula: see text] where [Formula: see text] is an arbitrary small positive constant and [Formula: see text] is the number of representations of [Formula: see text] as product of [Formula: see text] natural numbers.