Doubt in the simplification. I have the following equation where $\mathbf k=(k_1,\dots,k_n) \in \mathbb{Z}_+^n$ and $l|\mathbf k$ means that $l | k_i$ for each $i$.


Assume that $m_0(\eta(-))$, $m_1(\eta(-))$ and $|\pi_{\mathbf k}^G(q)[q]|$ are functions from $\mathbb{Z}_+^n$ to $\mathbb N$. I want find $m_0(\eta(-))$, $m_1(\eta(-))$ interms of  $\pi_{\mathbf k}^G(q)$ using Mobius inversion formula.
Kindly share your thoughts.
Thank you.
PS: Miguel has given a nice answer. But I think using Mobius inversion formula the resultant expression will be much simpler. This is my intuition but couldn't work out the details. For the first equation, I got the answer

I guess, for the second equation the answer will be $m_1(\eta(\textbf k))$ is equal to
. 
is it correct?
 A: Suppose $g.c.d(k_{1},...,k_{n})=1$, then $$m_{1}(\eta (k)) = |\pi_{k}^G(q)[q]|$$
Now for example if $g.c.d(k_{1},...,k_{n})=2$, we have $$m_{1}(\eta (k))-\frac{1}{2}m_{1}(\eta (\frac{k}{2}))=m_{1}(\eta (k))-\frac{1}{2}|\pi_{\frac{k}{2}}^G(q)[q]|$$ therefore $m_{1}(\eta (k))=\frac{1}{2}|\pi_{\frac{k}{2}}^G(q)[q]|+|\pi_{k}^G(q)[q]|$.
Let $d=g.c.d(k_{1},...,k_{n})$. If $d$ is prime then $m_{1}(\eta (k))=\frac{(-1)^{d+2}}{d}|\pi_{\frac{k}{d}}^G(q)[q]|+|\pi_{k}^G(q)[q]|$. 
Since we know the formula for $d$ prime I believe it should be possible to induct a general formula but it becomes quite complicated for me.
Edit:
Let $d$ denote $gcd(k)$ and $d_i$ denote $gcd(\frac{k}{i})$. From back substituting one can get $$m_{1}(\eta (k))=|\pi_{k}^G(q)[q]|+\sum_{l|d}\frac{(-1)^{l+2}}{l}m_{1}(\eta (\frac{k}{l}))=|\pi_{k}^G(q)[q]|+\sum_{l|d}\frac{(-1)^{l+2}}{l}(|\pi_{\frac{k}{l}}^G(q)[q]|+\sum_{i|d_l}\frac{(-1)^{i+2}}{i}m_1(\eta(\frac{k}{l\cdot i})))=|\pi_{k}^G(q)[q]|+\sum_{l|d}\frac{(-1)^{l+2}}{l}(|\pi_{\frac{k}{l}}^G(q)[q]|+\sum_{i|d_l}\frac{(-1)^{i+2}}{i}(|\pi_{\frac{k}{l\cdot i}}^G(q)[q]|+\sum_{j|d_{i\cdot l}}\frac{(-1)^{j+2}}{j}m_{1}(\eta (\frac{k}{l \cdot i}))))$$
Note that we can keep substituting until we get $m_{1}(\eta (\frac{k}{d}))=|\pi_{\frac{k}{d}}^G(q)[q]|$
$$m_{1}(\eta (k))=|\pi_{k}^G(q)[q]|+\sum_{l|d}\frac{(-1)^{l+2}}{l}(|\pi_{\frac{k}{l}}^G(q)[q]|+\sum_{i|d_l}\frac{(-1)^{i+2}}{i}(|\pi_{\frac{k}{l\cdot i}}^G(q)[q]|+\sum_{j|d_{i\cdot l}}\frac{(-1)^{j+2}}{j}(...|\pi_{\frac{k}{d}}^G(q)[q]|))...)$$
