A Note about Iterated Arithmetic Functions

TL;DR

This paper investigates the properties of iterated multiplicative arithmetic functions, proving that the limit function obtained through iteration is completely multiplicative under certain conditions.

Contribution

It establishes that the limit of iterated functions, under specified conditions, is a completely multiplicative function, extending understanding of arithmetic function iteration.

Findings

The limit function H is completely multiplicative.

Conditions on f ensure the structure of the limit function.

The paper characterizes the behavior of iterated multiplicative functions.

Abstract

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $\alpha$, $f(p^{\alpha})<p^{\alpha}$ and $f(p)\vert f(p^{\alpha})$. Suppose also that any prime that divides $f(p^{\alpha})$ also divides $pf(p)$. Define $f(0)=0$, and let $H(n)=\displaystyle{\lim_{m\rightarrow\infty}f^m(n)}$, where $f^m$ denotes the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely multiplicative.