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Weakly multiplicative arithmetic functions and the normal growth of groups

  • Jan-Christoph Schlage-PuchtaEmail author
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Abstract

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or \(\log \)-uniformly continuous normal order is close to a function of the form \(n\mapsto n^c\). As an application we show that a finitely generated, residually finite, infinite group, whose normal growth has a non-decreasing or a \(\log \)-uniformly continuous normal order, is isomorphic to \((\mathbb {Z}, +)\).

Keywords

Multiplicative functions Characterizations of \(n^c\) Monotonicity Normal order Normal subgroup growth 

Mathematics Subject Classification

11A25 

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Universität RostockRostockGermany

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