An Erdős-Kac theorem for Smooth and Ultra-Smooth integers

Abstract

We prove an Erdős-Kac type of theorem for the set $S(x,y)=\{n\le x:p|n\Rightarrow p\le y\}$. If $\omega (n)$ is the number of prime factors of $n$, we prove that the distribution of $\omega (n)$ for $n\in S(x,y)$ is Gaussian for a certain range of $y$ using method of moments. The advantage of the present approach is that it recovers classical results for the range $u=o(\log \log x)$ where $u=\frac{\log x}{\log y}$, with a much simpler proof.