On some sums involving the integral part function

Denote by [Formula: see text], [Formula: see text] and [Formula: see text] the number of representations of [Formula: see text] as a product of [Formula: see text] natural numbers, the number of distinct prime factors of [Formula: see text] and the characteristic function of the square-free integers, respectively. Let [Formula: see text] be the integral part of real number [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] for [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is an arbitrarily small positive number. These improve the corresponding results of Bordellès.