There are a number of generalizations of the Möbius function out there, which can be found by Google. But I'd just like to know if anything has been said about this: For $k \geq 2$, $k \in \mathbb{Z}$, let $\zeta$ be a primitive $k$-th root of unity. Let $\mu_k(n) = 0$ if $n$ is not squarefree. For squarefree $n$, let $\mu_k(n) = \zeta^{\omega(n)}$. Thank you very much.

**Update:** A recent beautiful paper of Zhi-Wei Sun, On a pair of zeta functions, answers the question contained here very nicely!