The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field.
type: Item
instance of: scholarly article
title: The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field.
zbMATH Open document ID: 1153.11340
DOI: 10.1016/S0022-314X(03)00062-3
author: Norisato Kataoka
published in: Journal of Number Theory
publication date: 2003-08-07T00:00:00Z
review text: Let \(k\) be a real quadratic number field and \(\mathfrak o_k\), \(E\) the ring of integers and the group of units in \(k\), respectively. Denote by \(E_{\mathfrak p}\) the subgroup represented by elements of \(E\) of \((\mathfrak o_k/\mathfrak p)\times\) for a prime ideal \(\mathfrak p\) in \(k\). The author proves that for a given positive rational integer \(a\), the set of prime numbers \(p\) for which the residual index of \(E_{\mathfrak p}\) for \(\mathfrak p\) lying above \(p\) is equal to \(a\) has a natural density \(c\) under the Generalized Riemann Hypothesis and the estimate \(c\mathrm{Li}(x)+O(x\log\log x/\log^ 2x)\) for prime numbers less than \(x\). Moreover, the explicit formula for the density \(c\) and a necessary and sufficient condition for \(c=0\) are given. Previous results can be found in \textit{K. Masima}, RIMS Kokyuroku 1026, 156--166 (1998; Zbl 1016.11543), \textit{Y.-M. Chen} et al., Nagoya Math. J. 158, 167--184 (2000; Zbl 0984.11057), and the author, Proc. Japan Acad., Ser. A 77, No. 10, 161--163 (2001; Zbl 1002.11075).
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R47
zbMATH DE Number: 1961840
zbMATH Keywords: Quadratic fields
zbMATH Keywords: Density
reviewed by: Olaf Ninnemann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1961840