On the asymptotic behavior of some counting functions

Volume 102 / 2005

Maciej Radziejewski, Wolfgang A. Schmid Colloquium Mathematicum 102 (2005), 181-195 MSC: 11N64, 11R27, 20K01. DOI: 10.4064/cm102-2-2


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.


  • Maciej RadziejewskiFaculty of Mathematics
    and Computer Science
    Adam Mickiewicz University
    Umultowska 87
    61-614 Poznań, Poland
  • Wolfgang A. SchmidInstitute for Mathematics
    and Scientific Computing
    Heinrichstrasse 36
    8010 Graz, Austria

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