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 ${\Psi }_n(x)$.

arXiv comment14 pages, 1 Table (computed by Yves Gallot)

Cited by in corpus (3)

• Value distribution of cyclotomic polynomial coefficients
• Values of coefficients of cyclotomic polynomials II
• Coefficients of cyclotomic polynomials