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

Volume 163 / 2014

Maciej Radziejewski Acta Arithmetica 163 (2014), 179-198 MSC: Primary 11N25; Secondary 11N45. DOI: 10.4064/aa163-2-7

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

Authors

  • Maciej RadziejewskiFaculty of Mathematics and Computer Science
    Adam Mickiewicz University
    Umultowska 87
    61-614 Poznań, Poland
    e-mail

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