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!
The partial sums of $\mu_k(n)$ are estimated in a paper of Addison:
A Note on the Compositeness of Numbers, A. W. Addison, Proceedings of the American Mathematical Society, Vol. 8, No. 1 (Feb., 1957), pp. 151-154,
Article Stable URL: http://www.jstor.org/stable/2032831
Most of the paper treats $\zeta^{\Omega(n)}$, but the (slightly different) function you are interested in is discussed in the closing paragraph.
If you are really interested in the sums $\sum_{n\leq x} \mu(n)^2e^{2i\pi a\omega(n)/k}$ (where $a$ could be an arbitrary integer), you should probably just look at what the Delange-Selberg method gives you. (A good reference is Tenenbaum's book on analytic and probabilistic number theory; the versions there may take some time to parse, because they are explicit in every possible parameter, but the result is worth the effort, since you will be able to deal with many more multiplicative functions using the same tools...)
