Möbius function and the independence of a sequence of random variables which is either i.i.d with null expectation, the product of such vars, or zero

Question

I am trying a question in probability and analytic number theory, concerning random multiplicative and Möbius functions, and cannot see how to proceed. Any kind of contribution would be appreciated.

Let $\mathcal{P}$ denote the set of primes, and $(X_p)_{p\in\mathcal{P}}$ be i.i.d. random vars with $\mathbb{P}(X_p=1) = \frac{1}{2} = \mathbb{P}(X_p=-1)$.

Also, for $n$ not prime, define:

• $X_n = \prod_{i=1}^{k} X_{p_i}$ , if $n=p_1 \cdot \ldots \cdot p_k$ distinct primes;
• $X_n=0$ , if $n$ is not square-free.

I need to know whether $(X_n)_{n\in\mathbb{N}}$ is $k$-wise independent (for any $k ≥ 3$), or at least pairwise independent, as it does not seem to be mutually independent as a set (since it's multiplicative?)