On a sum involving the number of distinct prime factors function related to the integer part function

Abstract

In this paper, we obtain asymptotic formula on the sum \sum\limits_{n\leq x}\omega \left( \left\lfloor \frac{x}{n}\right\rfloor \right) , where \omega \left( n\right) denote the number of distinct prime divisors of n and \left\lfloor t\right\rfloor denotes the integer part of t.