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On the distribution of prime divisors in arithmetical semigroups. (English) Zbl 0964.11036

The analytic theory of arithmetical semigroups was developed by A. Geroldinger and J. Kaczorowski [Sémin. Théorie Nombres Bordx. 4, 199-238 (1992; Zbl 0780.11046)]. An arithmetical semigroup \(S\) is said to be of type \(\beta\) if \(L(1,\chi)=0\) holds for a certain nontrivial character \(\chi\) of the associated class group of \(S\). In the note under review the author shows among other results that for a given arbitrary finite abelian group \(G\) of even order there exist infinitely many arithmetical semigroups \(S\) of type \(\beta\) with class group \(G\) and with in some sense prescribed distribution of the generators of \(S\).

MSC:

11M41 Other Dirichlet series and zeta functions
20M14 Commutative semigroups

Citations:

Zbl 0780.11046
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