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Non-unique factorizations in orders of global fields. (English) Zbl 0812.11061
Let $$O$$ be an order in an algebraic field $$K$$. The authors obtain asymptotics for the number of principal ideals $$I\subset O$$ with $$[O:I] \leq x$$, the generator of which either has only factorizations of length not exceeding $$k$$ or has factorizations of at most $$k$$ lengths or has at most $$k$$ distinct factorizations. In all three cases this number turns out to be asymptotically equal to $$Cx(\log x)^{-a} (\log\log x)^ b$$ with suitable $$C>0$$, $$0\leq a\leq 1$$ and a nonnegative integer $$b$$, having combinatorial interpretation. Here $$C$$ depends on $$k$$ and $$O$$ but $$a$$, $$b$$ depend only on $$k$$ and the structure of the Picard group of $$O$$ and in the first case one has $$a=1$$. The obtained results are similar to those obtained earlier in the case of principal orders [A. Geroldinger, Acta Arith. 57, 365-373 (1991; Zbl 0729.11054); F. Halter-Koch, ibid. 62, 173-206 (1992; Zbl 0762.11041); F. Halter- Koch and W. Mueller, J. Reine Angew. Math. 421, 159-188 (1991; Zbl 0736.11064); J. Kaczorowski, Acta Arith. 43, 53-68 (1983; Zbl 0526.12006)]. The authors utilize an axiomatic approach, which allows them to obtain at the same time similar assertions for more general structures, including generalized Hilbert semigroups.

##### MSC:
 11R27 Units and factorization 11N45 Asymptotic results on counting functions for algebraic and topological structures 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20M14 Commutative semigroups
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