Section: An Application of the Möbius Inversion Formula

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.

  1. Proposition: Sum of Euler Function
  2. Proposition: Explicit Formula for the Euler Function

Thank you to the contributors under CC BY-SA 4.0!