We consider an axiomatically-defined class of arithmetical semigroups that we call simple L-semigroups. This class includes all generalized Hilbert semigroups, in particular the semigroup of non-zero integers in any algebraic number field. We show, for all positive integers k, that the counting function of the set of elements with at most k distinct factorization lengths in such a semigroup has oscillations of logarithmic frequency and size for some M>0. More generally, we show a result on oscillations of counting functions of a family of subsets of simple L-semigroups. As another application we obtain similar results for the set of positive (rational) integers and the set of ideals in a ring of algebraic integers without non-trivial divisors in a given arithmetic progression.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-2-7,
author = {Maciej Radziejewski},
title = {The asymptotic behaviour of the counting functions of $\Omega$-sets in arithmetical semigroups},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {179-198},
zbl = {1306.11076},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-2-7}
}
Maciej Radziejewski. The asymptotic behaviour of the counting functions of Ω-sets in arithmetical semigroups. Acta Arithmetica, Tome 166 (2014) pp. 179-198. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-2-7/