The local distribution of the number of small prime factors: variation of the classical theme

Abstract

We obtain uniform estimates for $$N_k(x,y)$$, the number of positive integers n up to x for which $$\omega _y(n)=k$$, where $$\omega _y(n)$$ is the number of distinct prime factors of n which are $$ 0$$ by the Buchstab–de Bruijn method. We also utilize a certain recent result of Tenenbaum to complete our asymptotic analysis.