On the consecutive prime divisors of an integer

Abstract

Paul Erdös, Janos Galambos and others have studied the relative size of the consecutive prime divisors of an integer. Here, we further extend this study by examining the distribution of the consecutive neighbour spacings between the prime divisors $p_1(n)<p_2(n)<\cdots <p_r(n)$ of a typical integer $n\ge 2$. In particular, setting $\gamma_j(n):=\log p_j(n)/\log p_{j+1}(n)$ for $j=1,2,\ldots,r-1$ and, for any $\lambda\in(0,1]$, introducing $U_\lambda(n):= \#\{j\in\{1,2,\ldots,r-1\}: \gamma_j(n)<\lambda\}$, we establish the mean value of $U_\lambda(n)$ and prove that $U_{\lambda}(n)/r \sim \lambda$ for almost all integers $n\ge 2$. We also examine the shifted prime version of these two results and study other related functions.