On some sums involving the integral part function

Abstract

Denote by $τ$ k (n), $ω$(n) and $μ$ 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 = $ω$, 2 $ω$ , $μ$ 2 , $τ$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $ε$ (x $θ$ f +$ε$) for x $\rightarrow$ $\infty$, where $θ$ $ω$ = 53 110 , $θ$ 2 $ω$ = 9 19 , $θ$ $μ$2 = 2 5 , $θ$ $τ$ k = 5k--1 10k--1 and $ε$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{è}s.