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

Acta Arithmetica (2014)

- Volume: 163, Issue: 2, page 179-198
- ISSN: 0065-1036

## Access Full Article

top## Abstract

top## How to cite

topMaciej Radziejewski. "The asymptotic behaviour of the counting functions of Ω-sets in arithmetical semigroups." Acta Arithmetica 163.2 (2014): 179-198. <http://eudml.org/doc/279425>.

@article{MaciejRadziejewski2014,

abstract = {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.},

author = {Maciej Radziejewski},

journal = {Acta Arithmetica},

keywords = {counting functions; zeta function; arithmetic progression of ideals; -semigroups},

language = {eng},

number = {2},

pages = {179-198},

title = {The asymptotic behaviour of the counting functions of Ω-sets in arithmetical semigroups},

url = {http://eudml.org/doc/279425},

volume = {163},

year = {2014},

}

TY - JOUR

AU - Maciej Radziejewski

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

JO - Acta Arithmetica

PY - 2014

VL - 163

IS - 2

SP - 179

EP - 198

AB - 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.

LA - eng

KW - counting functions; zeta function; arithmetic progression of ideals; -semigroups

UR - http://eudml.org/doc/279425

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.