On the gap distribution of prime factors

2021/07/05 by de la Bretèche, Régis, Tenenbaum, Gérald
#11N25 #11N35 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2107.02055

Abstract

Let \p_j(n)\_j=1^ω(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^-zlog₂n and variance \\rm e^-z-2z\rm e^-2z\log₂n, and we establish an asymptotic formula with remainder for all centered moments.

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