Size of regulators and consecutive square-free numbers
type: Item
instance of: scholarly article
title: Size of regulators and consecutive square-free numbers
zbMATH Open document ID: 1320.11026
author: Florent Jouve
author: Étienne Fouvry
published in: Mathematische Zeitschrift
publication date: 2013-03-28T00:00:00Z
review text: Let \(\varepsilon_D=T+U\sqrt D>1\) be the fundamental solution to the Pell equation \(T^2-DU^2=1\) and let \[ S^f(x;\alpha):=\sharp \{D; \; D\leq x, D\neq \square, \varepsilon_D \leq D^{1/2+\alpha} \}. \] \textit{C. Hooley} [J. Reine Angew. Math. 353, 98--131 (1984; Zbl 0539.10019)] conjectured that there exist \(\delta(\alpha)\) and \(C(\alpha)\) such that \[ S^f(x;\alpha)\leq C(\alpha)x^{1-\delta(\alpha)} \] for every \(x\geq 2\). The main result of the paper under review is to show that for every \(\varepsilon >0\) \[ S^f(x;\alpha)\leq \mathcal O_\varepsilon\left(x^{\alpha/3+7/12+\varepsilon}\right) \] uniformly for \(\alpha\geq 0\) and \(x\geq 2\). The proof of their result relies on the square sieve introduced by \textit{D. R. Heath-Brown} [Math. Ann. 266, 251--259 (1984; Zbl 0514.10038)].
reviewed by: Volker Ziegler
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 6149074
zbMATH Keywords: quadratic fields
zbMATH Keywords: Pell equation
zbMATH Keywords: regulators
zbMATH Keywords: square sieve
MaRDI profile type: Publication
full work available at URL: s00209-012-1035-7
OpenAlex ID: W2143551211
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DOI: 10.1007/S00209-012-1035-7
description: scientific article; zbMATH DE number 6149074