On fractional sums of the divisor functions.

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 Theory18(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 S τ k (x) = ∑ n ≤ x τ k x n = ∑ n = 1 ∞ τ k (n) n (n + 1) x + O (x 9 / 1 9 + ) , where is an arbitrary small positive constant and τ k (n) is the number of representations of n as product of k natural numbers. [ABSTRACT FROM AUTHOR]