In answering a question here, I had to make use of the unique polynomial $P_N(x) = \sum_{j=0}^{N-1} a_j x^j \in \mathbb{C}[x]$ whose value at any primitive $N^{th}$ root of unity is $1$, and whose value at non-primitive $N^{th}$ roots of unity is $0$.
It occurred to me that, since this is the first time I've encountered this polynomial, I didn't know whether it has been particularly studied or has a nice form. I vaguely surmised it might be expressible in terms of cyclotomic polynomials, but so far haven't come up with much.
Some very partial results are
If $N = 2^m$, then $P_N(x) = \frac1{2}(1 - x^{N/2})$.
If $N = p$ is prime, then $P_N(x) = 1 - \frac1{p}(1 + x + \ldots + x^{p-1})$.
In these cases we have $P_N(x) = 1 - \frac{\Phi_N(x)}{\Phi_N(1)}$ where $\Phi_N(x)$ is the $N^{th}$ cyclotomic polynomial. But this pattern does not persist (it fails for $N = 6$).
Maybe some clever application of the inclusion-exclusion principle or the Möbius function has to get into the game, but it's a little murky to me.
