Sum of Powers of Primitive Complex Roots of Unity

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $U_n$ denote the complex $n$th roots of unity:
 * $U_n = \set {z \in \C: z^n = 1}$

Let $\alpha = \exp \paren {\dfrac {2 k \pi i} n}$ denote a primitive complex $n$th root of unity.

Let $s \in \Z_{>0}$ be a (strictly) positive integer.

Then:

where:
 * $n \divides s$ denotes that $n$ is a divisor of $s$
 * $n \nmid s$ denotes that $n$ is not a divisor of $s$.

Proof
First we address the case where $n \divides s$.

Then:

Hence:

Now let $n \nmid S$.

Then:

Hence: