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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→∞ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and w(n)/n are [0,1] and [0,∞), respectively. The same conclusions hold when n is restricted either to p2q with distinct primes p,q or to squarefree products of three primes. Nevertheless, on each of these three sets, the proportion of integers n≤ X satisfying P(w(n))/P(n)>α tends to zero as X→∞, for every fixed α>0. We also determine the complete set of finite joint limit points of the two quotients for n=p2q with p≠ q.
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BibTeX
@misc{yang2026density,
title = {{Density and distribution of quotients associated with Goldring's $w$-function}},
author = {Quan-Hui Yang and Jun-Jia Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02851},
eprint = {2611.02851},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02851},
}
RIS
TY - UNPB
TI - Density and distribution of quotients associated with Goldring's w-function
AU - Quan-Hui Yang
AU - Jun-Jia Zhao
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02851
UR - https://arxiv.org/abs/2611.02851
AB - 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→∞ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and w(n)/n are [0,1] and [0,∞), 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≤ X satisfying P(w(n))/P(n)>α tends to zero as X→∞, 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≠ q.
ER -