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On the asymptotic behavior of some counting functions. (English) Zbl 1143.11346
Summary: The investigation of certain counting functions of elements with given factorization properties in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group. In this paper a constant arising from the investigation of the number of algebraic integers with factorizations of at most $$k$$ different lengths is investigated. It is shown that this constant is positive if $$k$$ is greater than $$1$$ and that it is also positive if $$k$$ equals $$1$$ and the class group satisfies some additional conditions. These results imply that the corresponding counting function oscillates about its main term. Moreover, some new results on half-factorial sets are obtained.
Part II, Colloq. Math. 102, No. 2, 197–216 (2005; Zbl 1143.11351).

##### MSC:
 11N64 Other results on the distribution of values or the characterization of arithmetic functions 05A15 Exact enumeration problems, generating functions 11R27 Units and factorization 11R47 Other analytic theory 20K01 Finite abelian groups
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