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. Bordelles, 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 S-tau k (x) = sigma(n <= x) tau(k)([x/n]) = sigma(infinity)(n=1) tau(k)(n)/n(n + 1) x + O(x(9/19+epsilon)),where epsilon is an arbitrary small positive constant and tau(k)(n) is the number of representations of n as product of k natural numbers.