As a first application of the Möbius inversion, we want to calculate an explicit formula for the Euler function $\phi.$
As a first step, we have to find a function $f(n)$ with $$f(n)=\sum_{d\mid n}\phi(d).$$
Then, we will be able to apply the Möbius inversion formula, and get the sum $$\phi(n)=\sum_{d\mid n}\mu(n)f\left(\frac{n}{d}\right).$$
The last step will be to find an explicit formula for this sum.