On the number of prime factors of integers without large prime divisors (Q1090360)

Let \(\Omega\) (n) denote the total number of prime factors of n. The Erdős-Kac theorem asserts that the distribution of \(\Omega\) (n) on the set \(S(x)=\{n\in {\mathbb{N}}:\) \(n\leq x\}\) is essentially Gaussian with mean log log x and standard division \(\sqrt{\log \log x}\). The purpose of this paper is to prove an analogue of this theorem with S(x) replaced by \(S(x,y)=\{n\in {\mathbb{N}}:\) \(n\leq x\), p \(| n\Rightarrow p\leq y\}\). the author's result is an estimate with an error term which is uniform in the range log \(x\geq (\log y)^{21}\).