Irreducible algebraic integers in short intervals
type: Item
instance of: scholarly article
title: Irreducible algebraic integers in short intervals
zbMATH Open document ID: 1233.11114
author: J. Kaczorowski
published in: Mathematische Annalen
publication date: 2009-09-22T00:00:00Z
review text: The author continues his study [Monatsh. Math. 156, No. 1, 47--71 (2009; Zbl 1197.11146); Math. Nachr. 283, No. 9, 1291-1303 (2010; Zbl 1262.11092)] of the distribution of algebraic integers with certain factorization properties. In this paper he shows that if \(K\) is an algebraic number field with class-number \(\geq3\) and \(\theta>1/2\), then for \(x\geq x_0(K,\theta)\) (with effectively computable \(x_0(K,\theta)\)) there exists an irreducible element \(a\in K\) with \(x<|N(a)|<x+x^\theta\). He notes that this is stronger than the best known result \((\theta\geq0.525)\) obtained by \textit{R. C. Baker, G. Harman} and \textit{J. Pintz} [Proc. Lond. Math. Soc., III. Ser. 83, No. 3, 532--562 (2001; Zbl 1016.11037)] in the case of the field of rational numbers.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 5606237
zbMATH Keywords: number fields
zbMATH Keywords: factorizations
zbMATH Keywords: unique factorization
zbMATH Keywords: short intervals
MaRDI profile type: Publication
full work available at URL: s00208-009-0354-4
OpenAlex ID: W2064677895
cites work: The difference between consecutive primes. II
cites work: Q5200682
cites work: Some remarks on factorization in algebraic number fields
cites work: On the distribution of irreducible algebraic integers
cites work: \(\Omega \)-estimates related to irreducible algebraic integers
cites work: Topics in multiplicative number theory
cites work: Q4821147
cites work: Q3735790
DOI: 10.1007/S00208-009-0354-4
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description: scientific article; zbMATH DE number 5606237