# Sum of Powers of Euler's Totient Function.

I'm curious if there has been any work done to compute $$\sum_{d|x} \phi(d)^n$$ for integers $n>1$? I've been looking for a while and haven't found anything in the literature. I'm specifically looking to simplify or set lower bounds on this sum. Any help would be appreciated!

• It's necessarily multiplicative, so you just compute it for $x=p^k$ for some prime $p$. This is $$1+(p-1)^n+(p-1)^np^n +\cdots + (p-1)^np^{n(k-1)} = 1+(p-1)^n\frac{p^{nk}-1}{p^n-1}$$ – Thomas Andrews Nov 4 '16 at 15:21
• See OEIS sequence A029939 for $n=2$. $n=3$ doesn't seem to be in the OEIS. – Robert Israel Nov 4 '16 at 15:28
• I'm guessing the reason you haven't found much is that (1) it isn't hard, and (2) the function hasn't been needed anywhere significant. – Thomas Andrews Nov 4 '16 at 15:30
• @ThomasAndrews, thanks! And yeah I assume so as well. I was hoping there was a form that did not rely on the prime decomposition of x, but oh well. – j4l3kl24jkl2 Nov 4 '16 at 15:37