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An arithmetic function $f$ is multiplicative if $f(mn)=f(m)f(n)$ for coprime $m$, $n$. We say that an arithmetic function $g$ is completely multiplicative if:


...for all $m$, $n$. Let $\mathbb{P}$ be the set of all prime numbers.

Suppose that a multiplicative function $f$ satisfies the following property:

There is an integer $N$ and an arithmetic function $g$ such that for all $n\in \mathbb{N}$, we have:

$$ f(n)=n^N \prod_{p|n, \ p\in \mathbb{P}} g(p) $$

Find a necessary and sufficient condition for $f$ to be completely multiplicative.


$g(p)$ is $0$ or $1$ for any $p\in\mathbb{P}$.


$g$ is a constant function on $\mathbb{P}$, and the constant is $0$ or $1$.


$g(n)$ is $0$ or $1$ for any $n\in\mathbb{N}$.


$g$ is a multiplicative function.


$g$ is a completely multiplicative function.

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