On some sums involving the integral part function

Abstract

Denote by $\ddot{I}„$ k (n), $\ddot{I}‰$(n) and $\hat{I}¼$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $\ddot{I}‰$, 2 $\ddot{I}‰$ , $\hat{I}¼$ 2 , $\ddot{I}„$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $\hat{I}\mu$ (x $\hat{I}¸$ f +$\hat{I}\mu$) for x $\to$ $\infty$, where $\hat{I}¸$ $\ddot{I}‰$ = 53 110 , $\hat{I}¸$ 2 $\ddot{I}‰$ = 9 19 , $\hat{I}¸$ $\hat{I}¼$2 = 2 5 , $\hat{I}¸$ $\ddot{I}„$ k = 5k--1 10k--1 and $\hat{I}\mu$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{è}s.