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

Maciej 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 -