Note on distribution of units of real quadratic number fields

Abstract

Let $k$ be a real quadratic number field and $\mathfrak{o}_k$, $E$ the ring of integers and the group of units in $k$. Denote by $E_{\mathfrak{p}}$ a subgroup represented by $E$ of $(\mathfrak{o}_k / \mathfrak{p})^{\times}$ for a prime ideal $\mathfrak{p}$ in $k$. We report that for a given positive integer $a$, the set of prime ideals of degree 1 for which the residual index of $E_{\mathfrak{p}}$ is equal to $a$ has a density under the Generalized Riemann Hypothesis.