Let $\sigma $ act on the $m$th roots of unity by $\sigma \zeta = \zeta^r$ for $1\le r
I keep running into this pesky little sum and was wondering if anyone on this forum might be able to point me in the direction of converting it to something simple (or at least more amenable for further computation.) It wouldn't surprise me if I'm overlooking something elementary, but it's already frustrated me to the point of posting here, so any guidance whatsoever would be greatly appreciated.