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 \((\mathbb {Z}, +)\).
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Schlage-Puchta, JC. Weakly multiplicative arithmetic functions and the normal growth of groups. Arch. Math. 112, 233–240 (2019). https://doi.org/10.1007/s00013-018-1267-9
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DOI: https://doi.org/10.1007/s00013-018-1267-9