Let $n=p^{\alpha_1}_1 \cdots p^{\alpha_m}_m,$ and define $$\lambda_k(n)= (-1)^{ [\frac{\Omega(n)}{k} ]},$$ where $\Omega(n)= \alpha_1 + \cdots + \alpha_k,$ and $[\cdot]$ is the floor function.

For $k=1$, $\lambda_1$ is the Liouvilles Lambda function. For $k=2$:

$\lambda_2(1)=1, \hspace{2 mm} \lambda_2(p_1)=1, \hspace{2 mm} \lambda_2(p_1p_2)=-1, \hspace{2 mm} \lambda_2(p_1p_2p_3)=-1 \hspace{2 mm} \text{ and so on...}.$

-Is there anythig known about this function?

-If for $\Re(s)>1$ we define $$L(s, \lambda_2):= \sum_{n=1}^{\infty} \frac{\lambda_2(n)}{n^s},$$

Is there any connection between $L_{\lambda_2}(s)$ and the zeros of the Riemann zeta function?

-I guess (without knowing how to prove) that $$\sum_{n<x} \lambda_2(n) = o(x),$$ smilar to $\lambda_1$ should we expect square root cancelation on sum of $\lambda_2$ as well?