The asymptotic behaviour of the counting functions of Ω-sets in arithmetical semigroups

• Autorzy: Maciej Radziejewski
• Języki publikacji: EN

Abstrakty

EN

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 $√x(logx)^{-M}$ 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.

Kategorie tematyczne

• 11N25: Distribution of integers with specified multiplicative constraints
• 11N45: Asymptotic results on counting functions for algebraic and topological structures

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 2
• Strony: 179-198

Daty

wydano

2014

Twórcy

autor

Maciej Radziejewski

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

Identyfikatory

DOI

10.4064/aa163-2-7