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 $<y$. The motivation for this problem is an observation due to the first author in 2015 that for certain ranges of $y$, the asymptotic behavior of $N_k(x,y)$ is different from the classical situation concerning $N_k(x,x)$ studied by Sathe and Selberg. We demonstrate this variation of the classical theme; to estimate $N_k(x,y)$ we study the sum $S_z(x,y)=\sum_{n\le x}z^{\omega_y(n)}$ for $Re(z)\>0$ by the Buchstab-de Bruijn method. We also utilize a certain recent result of Tenenbaum to complete our asymptotic analysis.