Consecutive neighbour spacings between the prime divisors of an integer
Volume 170 / 2022
Abstract
Writing $p_1(n) \lt \cdots \lt p_r(n)$ for the distinct prime divisors of a given integer $n\ge 2$ and letting, for a fixed $\lambda \in (0,1]$, $U_\lambda (n):=\#\{j\in \{1,\dots ,r-1\}:\log p_j(n)/\log p_{j+1}(n) \lt \lambda \}$, we recently proved that $U_\lambda (n)/r \sim \lambda $ for almost all integers $n\ge 2$. Now, given $\lambda \in (0,1)$ and $p\in \wp $, the set of prime numbers, let ${\cal B}_\lambda (p):=\{q\in \wp : \lambda \lt \frac {\log q}{\log p} \lt 1/\lambda \}$ and consider the arithmetic function $u_\lambda (n):= \#\{p\,|\, n: (n/p,{\cal B}_\lambda (p))=1\}$. Here, we prove that $\sum _{n\le x} (u_\lambda (n) - \lambda ^2 \log \log n)^2 = (C+o(1))x\log \log x$ as $x\to \infty $, where $C$ is a positive constant which depends only on $\lambda $, and thereafter we consider the case of shifted primes. Finally, we study a new function $V(n)$ which counts the number of divisors of $n$ with large neighbour spacings and establish the mean value of $V(n)$ and of $V^2(n)$.
