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

100%
Radziejewski M.
Acta Arithmetica
|
2014
|
tom 163
|
nr 2
179-198
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.
2
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On the distribution of algebraic numbers with prescribed factorization properties

100%
Radziejewski M.
Acta Arithmetica
|
2005
|
tom 116
|
nr 2
153-171
3
Content available remote

A note on certain semigroups of algebraic numbers

100%
Radziejewski M.
Colloquium Mathematicum
|
2001
|
tom 90
|
nr 1
51-58
EN
The cross number κ(a) can be defined for any element a of a Krull monoid. The property κ(a) = 1 is important in the study of algebraic numbers with factorizations of distinct lengths. The arithmetic meaning of the weaker property, κ(a) ∈ ℤ, is still unknown, but it does define a semigroup which may be interesting in its own right. This paper studies some arithmetic(divisor theory) and analytic(distribution of elements with a given norm) properties of that semigroup and a related semigroup of ideals.
4
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On the asymptotic behavior of some counting functions

63%
Radziejewski M., Schmid W.
Colloquium Mathematicum
|
2005
|
tom 102
|
nr 2
181-195
EN
The investigation of certain counting functions of elements with given factorization properties in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group. In this paper a constant arising from the investigation of the number of algebraic integers with factorizations of at most k different lengths is investigated. It is shown that this constant is positive if k is greater than 1 and that it is also positive if k equals 1 and the class group satisfies some additional conditions. These results imply that the corresponding counting function oscillates about its main term. Moreover, some new results on half-factorial sets are obtained.
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