Does anybody know how to find a nice upper bound for a sum of the type
$$\displaystyle\sum_{k\mid n\atop{ k\le x}}\varphi(k)$$ when $n$ is fixed?
I found that this sum is close to $x$ when $n$ is a power of a prime.
Note: $\varphi(\cdot)$ is the Totient Euler Function. Thks.