On the gap distribution of prime factors

Abstract

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. For $z\geqslant 0$, let $G(n;z)$ denote the number of those indexes $j$ such that $p_{j+1}(n)>p_j(n)^{\exp z}$. We show uniform convergence, with almost optimal effective estimate of the speed, of the distribution of $G(n;z)$ on $\{n:1\leqslant n\leqslant N\}$ to a Gaussian limit law with mean ${\rm e}^{-z}\log_2n$ and variance $\{{\rm e}^{-z}-2z{\rm e}^{-2z}\}\log_2n$, and we establish an asymptotic formula with remainder for all centered moments.