Density and distribution of quotients associated with Goldring's $w$-function

Abstract

For primes $p,q,r$ that are not all equal, let $n=pqr$ and define Goldring's $w$-function by $w(n)=P(p+q)P(p+r)P(q+r)$, where $P(a)$ denotes the largest prime factor of $a$. In this paper, we resolve three problems of Chen, Shi and Wu. As $n\to\infty$ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and $w(n)/n$ are $[0,1]$ and $[0,\infty)$, respectively. The same conclusions hold when $n$ is restricted either to $p^2q$ with distinct primes $p,q$ or to squarefree products of three primes. Nevertheless, on each of these three sets, the proportion of integers $n\le X$ satisfying $P(w(n))/P(n)>α$ tends to zero as $X\to\infty$, for every fixed $α>0$. We also determine the complete set of finite joint limit points of the two quotients for $n=p^2q$ with $p\ne q$.