I am having a little trouble with this question.
Given an arithmetic function f, a “Dirichlet square root” of f is an arithmetic function g such that $g ∗ g = f$. Prove by elementary techniques that the constant function $1$ has two Dirichlet square roots, of the form $±g$, where $g$ is a multiplicative function, and find the values of $g$ at prime powers.
I am trying to solve it using strong induction.
Since $$g(1)g(1)=1$$
we can define $g(1)=1$ and supposing $g(1), g(2), \dots, g(n-1)$ are well-defined, we must have $$1 = 2g(1)g(n)+\Sigma_{d|n,1<d<n}g(d)g(n/d)$$
and so we can define $g(n)$.
It is easy to prove that $-g$ is also a square root of the unity, providing that $g$ is defined.
However, I am not sure if this "proof" is correct. It seems weird to me. And I don't know how to prove that $g$ is multiplicative.