On some sums involving the integral part function

Denote by $\tau$ k (n), $\omega$(n) and $\mu$ 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 = $\omega$, 2 $\omega$ , $\mu$ 2 , $\tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $\epsilon$ (x $\theta$ f +$\epsilon$) for x $\rightarrow$ $\infty$, where $\theta$ $\omega$ = 53 110 , $\theta$ 2 $\omega$ = 9 19 , $\theta$ $\mu$2 = 2 5 , $\theta$ $\tau$ k = 5k--1 10k--1 and $\epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{\`e}s.