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

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 √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. © Instytut Matematyczny PAN 2014.