201803.08964v1 The local distribution of the number of small prime factors - variation of the classical theme

We obtain uniform estimates for Nk(x,y), the number of positive integers n up to x for which ωy(n)=k, where ωy(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 Nk(x,y) is different from the classical sit..