On the distribution of Ω(n)−ω(n)

Abstract

Let $\Omega \left(n\right)$ and $\omega \left(n\right)$ be the number of all prime factors and distinct prime factors of n, respectively. In 1955, Rényi found the density for the numbers n such that $\Omega \left(n\right)-\omega \left(n\right)=k$ for all integers $k\ge 0$. In this paper, we generalize Rényi's theorem and give a short and elementary proof. Moreover, we show that the distribution of $\Omega \left(n\right)-\omega \left(n\right)$ displays a disjoint form with the arithmetic functions of invariant averages under multiplications. As a consequence, we obtain some ergodic theorems on $\Omega \left(n\right)-\omega \left(n\right)$ that build connections among Rényi's theorem, the prime number theorem, Bergelson-Richter's theorem and Loyd's theorem.