Weakly multiplicative arithmetic functions and the normal growth of groups

TL;DR

This paper demonstrates that certain weakly multiplicative arithmetic functions with a regular normal order are close to power functions, and applies this to show that specific groups with regular normal growth are isomorphic to the integers.

Contribution

It establishes a link between weak multiplicativity and power-like behavior of functions, and characterizes groups with regular normal growth as infinite cyclic groups.

Findings

Arithmetic functions with weak multiplicativity and regular normal order resemble power functions.

Finitely generated, residually finite groups with regular normal growth are isomorphic to the integers.

Normal growth conditions imply the group is isomorphic to (Z, +).

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 $(\mathbb{Z}, +)$.