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 $z⩾0$, 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 $\{n:1⩽n⩽N\}$ to a Gaussian limit law with mean ${\mathrm{e}}^{-z}{\log }_{2}n$ and variance $\{{\mathrm{e}}^{-z}-2z{\mathrm{e}}^{-2z}\}{\log }_{2}n$, and we establish an asymptotic formula with remainder for all centered moments.