Relative block semigroups and their arithmetical applications

Franz Halter-Koch

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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 $\sqrt x {(l o g x)}^{- M}$ for some M>0. More generally, we show...