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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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