On the asymptotic behavior of some counting functions
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.