Do these multiplicative functions exist?

Question

I read about the multiplicative function $\mu$ defined by Möbius. Which is for any given $n\in \mathbb{N}$ such as $n=p_1^{\alpha_1}\cdot ....\cdot p_r^{\alpha_r}$ is defined as $$\mu(n)= \left\{ \begin{array}{lcc} 0 & \text{ if there is } 1\leq i \leq r \text{ such as } \alpha_i\geq 2 \\ 1 & \text{ if } n=1 \\ (-1)^r & \text{ any other case } \end{array} \right.$$ Now this raises the question that if these functions have already been defined.
For any fixated $k\in \mathbb{N}$ and the same $n\in \mathbb{N}$ as before, we define. $$\mu_k(n)= \left\{ \begin{array}{lcc} 0 & \text{ if there is } 1\leq i \leq r \text{ such as } \alpha_i\geq k \\ 1 & \text{ if } n=1 \\ e^{\frac{2 \pi i m}{k}} & \text{ any other case.} \end{array} \right.$$ Being $m=\Omega(n)=\displaystyle\sum_{i=1}^r \alpha_i$
It would make sense that this is the case, since these functions are all defined in the roots of unity, which is similar to the original $\mu$ and also multiplicative, but I can't seem to find them anywhere. Can anyone tell me if I'm correct?

Answer 1

Note $u_1(n)=|u_k(n)|=|\mu(n)|$ (see OEIS Entry A008966), and $u_2(n)=\mu(n)$ (see OEIS Entry A008683).

With respect to $u_k(n)$ more generally, the Dirichlet characters $\chi_{k,j}(n)$ are of much greater theoretical interest (see Wikipedia: Dirichlet Characters and MathWorld: Number Theoretic Character).