Reciprocal cyclotomic polynomials

Abstract

Let $\Psi_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $\Psi_n(x)=(x^n-1)/\Phi_n(x)$, with $\Phi_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $\Psi_n(x)$ are integers that like the coefficients of $\Phi_n(x)$ tend to be surprisingly small in absolute value, e.g. for $n<561$ all coefficients of $\Psi_n(x)$ are $\le 1$ in absolute value. We establish various properties of the coefficients of $\Psi_n(x)$.