Irreducible algebraic integers in short intervals
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.
scientific article; zbMATH DE number 5606237
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