Let $n$ be an integer greater $2$. What is the value of the following product for $a=3, \dots , n-1$:
$$\prod_{k = 1}^{n-1} \left( 1 - \sum_{j = 1}^{a-1} \zeta^{jk} \right)$$
where $\zeta$ is some complex $n$th root of unity. I can prove that:
$$\prod_{k = 1}^{n-1} \left( 1 + \sum_{j = 1}^{a-1} \zeta^{jk} \right) = 1$$
and for $a = 2$:
$$\prod_{k = 1}^{n-1} \left( 1 - \sum_{j = 1}^{a-1} \zeta^{jk} \right) = \prod_{k = 1}^{n-1} (1-\zeta^k) = n$$
However, I have problem with the general case.