# Math Help - Euler function phi(n) using the Moubilus function

1. ## Euler function phi(n) using the Moubilus function

Hi,
How can I do that?
Express Euler function phi(n) using the Moubilus function miu(n)
Thank you!

2. Originally Posted by bamby
Hi,
How can I do that?
Express Euler function phi(n) using the Moubilus function miu(n)
Thank you!
Consider the group $\mathbb{Z}_n$. Every element has order a divisor of $n$. Also if $d|n$ then the number of elements that has order $d$ is $\phi (d)$. Therefore, we have that $\sum_{d|n}\phi (d) = n$.

Since this is true for all $n\geq 1$ by Mobius inversion formula we see that, $\phi (n) = n\left( \sum_{d|n} \frac{\mu (d)}{d} \right)$.