On the asymptotic behavior of some counting functions

Maciej Radziejewski¹ ; Wolfgang A. Schmid²

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland
² Institute for Mathematics and Scientific Computing Karl-Franzens-Universität Heinrichstrasse 36 8010 Graz, Austria

Colloquium Mathematicum, Tome 102 (2005) no. 2, pp. 181-195

Résumé

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.

Détail

DOI : 10.4064/cm102-2-2

Keywords: investigation certain counting functions elements given factorization properties ring integers algebraic number field gives rise combinatorial problems class group paper constant arising investigation number algebraic integers factorizations different lengths investigated shown constant positive greater positive equals class group satisfies additional conditions these results imply corresponding counting function oscillates about its main term moreover results half factorial sets obtained