I am well aware of the Möbius inversion formula, which states $$\sum_{d \mid q} \mu(q/d) = \mathbf{1}_{q=1}$$
Do we know how to give a closed formula for the "partial" such convolution $$\sum_{a \mid d \mid q} \mu(q/d) = \sum_{d \mid q/a} \mu(q/d) = ?$$
Write $d=at$, so we have
$$ S=\sum_{\substack{t\in\mathbb Z\\at|q}}\mu\left(q\over at\right)=\sum_{t|(q/a)}\mu\left(q/a\over t\right)=\mathbf{1}_{q=a}. $$