Usage of $n=∑_{d|n}Φ(d)$

Question

Let Φ be Euler's totient function. Please tell me the usage of $n=∑_{d|n}Φ(d)$. Why is this formula important ? One of the usage I found is to use this formula to derivate some another formula to calculate $∑_{i=1}^{n}Φ(i) $ in O($n^{2/3}\log{n})$ . Is there any other usage?

Answer 1

This has applications for arithmetic functions and the Dirichlet product $f\ast g$, in analytic number theory. Then the identity is equivalent to $$ id=\epsilon\ast \phi, $$ which implies that $$ \phi=\epsilon^{-1}\ast id=\mu\ast id, $$ which translates to $$ \phi(n)=\sum_{d\mid n}\mu(d)\frac{n}{d}, $$ with the Moebius $\mu$-function. This can be applied to zeta functions and results about prime number distribution. Another consequence is an identity by Selberg, which is the starting point for the so-called elementary proof of the Prime Number Theorem.

Answer 2

Well, it is nice, isn't it? But it also appears in the proof of the claim that any finite subgroup of the multiplicative group of a field is cyclic...

You can check this in the first few pages of the wonderful little book "A Course in Arithmetic", by J.P. Serre