On the set of divisors of Gaussian integers
\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 \[ |\log |\pi'/\pi|| \leq \Delta_r(n) \log n \quad \text{and} \quad |\arg(\pi'/\pi)| \leq \Delta_{\arg}(n). \]
scientific article; zbMATH DE number 7140968
scientific article