Problem

Source: China TST 2021, Test 2, Day 2 P4

Tags: number theory, totient function, function



Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.