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

We obtain uniform estimates for N k (x , y) , the number of positive integers n up to x for which ω y (n) = k , where ω 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) = ∑ n ≤ x z ω y (n) for R e (z) > 0 by the Buchstab–de Bruijn method. We also utilize a certain recent result of Tenenbaum to complete our asymptotic analysis. [ABSTRACT FROM AUTHOR]