On the gap distribution of prime factors

Abstract

Let{pj(n)}j=1ω(n)\{p_j(n)\}_{j=1}^{\omega (n)}denote the increasing sequence of distinct prime factors of an integer nn. Forz⩾0z\geqslant 0, letG(n;z)G(n;z)denote the number of those indexesjjsuch thatpj+1(n)>pj(n)exp⁡zp_{j+1}(n)>p_j(n)^{\operatorname {exp} z}. We show uniform convergence, with almost optimal effective estimate of the speed, of the distribution ofG(n;z)G(n;z)on{n:1⩽n⩽N}\{n:1\leqslant n\leqslant N\}to a Gaussian limit law with meane−z⁡log2⁡n\operatorname {e}^{-z}\log _2n, variance{e−z−2ze−2z}log2⁡n\{\operatorname {e}^{-z}-2z\operatorname {e}^{-2z}\}\log _2n, and we establish an asymptotic formula with remainder for all centered moments.