Sum of equidistant exponents
For me, only the case of an even number of members of the sum is clear. How to prove this in general terms?

ah, &quot;sum orthogonality proof&quot; is something you'll want to google. You can do it (under a relaxed meaning of &quot;proof&quot;) very geometrically. Understand each part in your sum as vector in a plane:

take the first vector: $e^{0}\equiv 1$. It goes from point 0+0j to 1+0j.
attach the second vector to the tip of that vector. Note the inner angle between these two. Also note how the position of the tip is the partial sum of the first two elements from your sum.
attach the third to the tip of the second. Note that the angle between these two is the same.
Repeat until you're out of vectors.

You clearly followed a polygonal chain, where each of the $N$ edges had length 1. Now, what is the condition on the angles for such a polygonal chain to be a closed polygon?