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arXiv · 1912.00851

Weakly multiplicative arithmetic functions and the normal growth of groups

Jan-Christoph Schlage-Puchta
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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 $(\mathbb{Z}, +)$.

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Jan-Christoph Schlage-Puchta. 2019-12-02. Weakly multiplicative arithmetic functions and the normal growth of groups. https://doi.org/10.1007/s00013-018-1267-9

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