An Erd\H{o}s-Kac theorem for Smooth and Ultra-Smooth integers

TL;DR

This paper establishes a Gaussian distribution law for the number of prime factors of smooth and ultra-smooth integers within a certain range, extending classical results with a simpler proof using the method of moments.

Contribution

It proves an Erd ext{o}s-Kac type theorem for smooth integers, providing a new, simpler proof that covers a broader range of parameters than previous results.

Findings

Distribution of prime factors is Gaussian for smooth integers in specified range

Method of moments effectively proves the theorem

Recovers classical results with simpler proof

Abstract

We prove an Erd\H{o}s-Kac type of theorem for the set $S(x,y)=\{n\leq x: p|n \Rightarrow p\leq y \}$. 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.