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A new invariant for tame abelian extensions. (English) Zbl 0665.12004
Suppose \(K| \mathbb Q\) is tame and abelian with \(\Gamma =\text{Gal}(K| \mathbb Q)\). Let \(\mathfrak O_ K\) denote the ring of algebraic integers in \(K\). Then a basis of \(\mathfrak O_ K\) exists (over \(\mathbb Z\)), consisting of the conjugates of a single element, that is \({\mathfrak O}_ K=a.\mathbb Z\Gamma\) for some \(a\in {\mathfrak O}_ K\). The set of all such generators is the orbit \(a.\mathbb Z\Gamma^*\), \(\mathbb Z\Gamma^*\) denoting the units of \(\mathbb Z\Gamma\).
In the paper the aim is to “count” these generators with respect to the norm map \(N_ K| \mathbb Q: K\to \mathbb Q\). This is achieved by studying the Dirichlet series \[ \sum_{x\in\mathbb Z\Gamma^*}(\log | N_{K| \mathbb Q}(a,x)|)^{-s},\quad s\in\mathbb C. \] The series is shown to define a meromorphic function in the half-plane \(\text{Re}(s)>r_{\Gamma}-2\) (where \(r_{\Gamma}\) denotes the torsion free rank of \(\mathbb Z\Gamma^*)\). Here it is analytic apart from simple poles at \(s=r_{\Gamma}\), \(r_{\Gamma}-1\). Explicit, closed formulae for the residues are given. The residue at \(s=n\) is independent of \(K\), that at \(s=r_{\Gamma}-1\) contains a factor \(\log m_ K\), where \(m_ K\) is a positive integer which is a sort of “discriminantal” constant, an invariant of \(K\).
Result of this kind imply results for the counting function: \[ N_ K(q)=\#\{x\in \mathbb Z\Gamma^*: | N_{K|\mathbb Q}(a,x)| <q). \] Namely, \[ N_ K(q)=\alpha (\log q)^{r_{\Gamma}}+\beta (\log q)^{r_{\Gamma}-1}+O((\log q)^{r_{\Gamma}-1}),\quad q\to \infty. \] The Dirichlet series plays an essential role in the explicit determination of the constants \(\alpha\) and \(\beta\).

MSC:
11R21 Other number fields
11R04 Algebraic numbers; rings of algebraic integers
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