The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field.
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).
scientific article; zbMATH DE number 1961840