On the set of divisors of Gaussian integers
type: Item
instance of: scholarly article
title: On the set of divisors of Gaussian integers
zbMATH Open document ID: 1480.11116
author: Helmut Maier
author: Saurabh Kumar Singh
published in: The Ramanujan Journal
publication date: 2019-12-06T00:00:00Z
review text: \textit{H. Maier} and \textit{G. Tenenbaum} [Invent. Math. 76, 121--128 (1984; Zbl 0536.10039)] proved a conjecture of Erdős saying that almost all integers have a pair of divisors \(d, d'\) such that \(d < d' \leq 2d\). More precisely, they proved that, for almost all \(n\), one has \[ \inf_{d, d' \mid n} \left| \log \frac{d'}{d}\right| \leq \frac{\exp(\xi(n) \sqrt{\log \log n})}{(\log n)^{\log 3-1}} \] for any \(\xi(n)\) tending to infinity with \(n\). In the current paper, the authors consider a similar question for Gaussian integers -- they prove that almost all Gaussian integers have two divisors that are close to each other. They use a method similar to Maier and Tenenbaum, but there are additional complications coming from the two-dimensionality of the problem since now both the absolute values and the arguments of the divisors are need to be close to each other. More precisely, the authors show that if \(\Delta_r(n)\) and \(\Delta_{\arg}(n)\) are slowly varying and such that \(\Delta_r(n)^2 \Delta_{\arg}(n) \geq (\log n)^{-\log 3 + \varepsilon}\), then, for almost all Gaussian integers \(n\) there exist a pair of divisors \(\pi, \pi' \mid n\) (with \((\pi, \pi') = 1\)) such that \[ |\log |\pi'/\pi|| \leq \Delta_r(n) \log n \quad \text{and} \quad |\arg(\pi'/\pi)| \leq \Delta_{\arg}(n). \]
reviewed by: Kaisa Matomäki
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 7140968
zbMATH Keywords: Gaussian integers
zbMATH Keywords: divisors
zbMATH Keywords: Erdős conjecture on divisors
MaRDI profile type: Publication
full work available at URL: s11139-019-00158-9
OpenAlex ID: W2969575484
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DOI: 10.1007/S11139-019-00158-9
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description: scientific article; zbMATH DE number 7140968