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The conductor density of abelian number fields. (English) Zbl 0727.11041
Let G be a finite abelian group, and let K range over all absolutely abelian number fields with Galois group Gal(K/$${\mathbb{Q}})\cong G$$. Let $${\mathcal N}(x)=\sum_{0<f\leq x}N(f)$$, where N(f) is the number of fields K with conductor equal to f. The main results are: (i) An explicit formula for N(f). (ii) An asymptotic formula of the type $${\mathcal N}(x)=x P(\log x)+O(x^{1-\delta +\epsilon}),$$ where P is a polynomial and $$\delta$$ is a given positive constant. Explicit expressions for the degree and the leading coefficient of P are computed. The results should be compared with those of M. J. Taylor [J. Lond. Math. Soc., II. Ser. 29, 211- 223 (1984; Zbl 0535.12009)] who considered the case when G is cyclic and the basefield contains certain roots of unity.
Reviewer: S.Mäki (Turku)

##### MSC:
 11R20 Other abelian and metabelian extensions 11N45 Asymptotic results on counting functions for algebraic and topological structures
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