A Note about Iterated Arithmetic Functions

Abstract

Let $f:\mathbb{N}\to {\mathbb{N}}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $\hat{I}\pm$, $f(p^{\hat{I}}\pm )<p^{\hat{I}}\pm$ and $f(p)|f(p^{\hat{I}}\pm )$. Suppose also that any prime that divides $f(p^{\hat{I}}\pm )$ also divides $pf(p)$. Define $f(0)=0$, and let $H(n)=\underset{m\to \infty }{\lim }{f}^m(n)$, where $f^m$ denotes the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely multiplicative.

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