On the distribution of irreducible algebraic integers
type: Item
instance of: scholarly article
title: On the distribution of irreducible algebraic integers
zbMATH Open document ID: 1197.11146
author: J. Kaczorowski
published in: Monatshefte für Mathematik
publication date: 2009-03-24T00:00:00Z
review text: Let \(K\) be an algebraic number field with class-number \(h\geq2\), let \(M_K\) be the set of all principal ideals of \(K\), generated by irreducible elements, let \[ \zeta(s,M_K)=\sum_{I\in M_K}{1\over N(I)^s} \] be the corresponding zeta-function and put \[ T_K(x)={1\over2\pi i}\int_{\Gamma}\zeta(s,M_K){x^s\over s}ds, \] where \(\Gamma\) is a suitable curve surrounding the pole of \(\zeta(s,M_K)\) at \(s=1\). It follows from earlier results of the author [Acta Arith. 43, 53--68 (1983; Zbl 0526.12006)] that for the counting function \(M_K(x)\) of \(M_K\) one has \[ M_K(x)=T_K(x)+E_K(x) \] with \[ E_K(x)=O\left(x\exp\left(-a_K\sqrt{\log x}\right)\right). \] The author shows under certain assumptions about the multiplicity of zeros of Hecke \(L\)-functions corresponding to \(K\) that the error term \(E_K(x)\) has oscillations of logarithmic frequency and size \[ \sqrt x(\log\log x)^{D(K)-1}/\log x, \] \(D(K)\) denoting the constant of Davenport of the class-group of \(K\). He shows also that the assumption about \(L\)-functions is satisfied if either \(h=2\) or \(K\) is of prime degree \(p\) and the Dedekind zeta function of the Hilbert class-field of \(K\) has at least one non-trivial zero of multiplicity not divisible by \(p\).
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11N64
Mathematics Subject Classification ID: 11R47
zbMATH DE Number: 5532497
zbMATH Keywords: factorizations in number fields
zbMATH Keywords: irreducible algebraic integers
zbMATH Keywords: omega theorems
zbMATH Keywords: sign changes
zbMATH Keywords: oscillation of error terms
MaRDI profile type: Publication
full work available at URL: s00605-008-0559-8
OpenAlex ID: W2025576149
cites work: Oscillatory properties of arithmetical functions. II
cites work: Functional independence of the singularities of a class of Dirichlet series
cites work: Almost periodicity of some error terms in prime number theory
cites work: Ω-estimates for a class of arithmetic error terms
cites work: Oscillations of a given size of some arithmetic error terms
DOI: 10.1007/S00605-008-0559-8
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description: scientific article; zbMATH DE number 5532497