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.