The asymptotic behaviour of the counting functions of $\varOmega $-sets in arithmetical semigroups

Maciej Radziejewski¹

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland

Acta Arithmetica, Tome 163 (2014) no. 2, pp. 179-198

Résumé

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 $\sqrt {x}(\log x)^{-M}$ 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.

Détail

DOI : 10.4064/aa163-2-7

Keywords: consider axiomatically defined class arithmetical semigroups call simple l semigroups class includes generalized hilbert semigroups particular semigroup non zero integers algebraic number field positive integers counting function set elements distinct factorization lengths semigroup has oscillations logarithmic frequency size sqrt log m generally result oscillations counting functions family subsets simple l semigroups another application obtain similar results set positive rational integers set ideals ring algebraic integers without non trivial divisors given arithmetic progression