Zeros and irreducibility of polynomials with gcd powers as coefficients

Abstract

We study the family of self-inversive polynomials of degree \(n\), whose \(j\)th coefficient is \(\gcd (n,j)^k\), for each fixed integer \(k \ge 1\). We prove that these polynomials have all of their roots on the unit circle, with uniform angular distribution. In the process we prove some apparently new results on Jordan’s totient function. We also prove that these polynomials are irreducible, apart from an obvious linear factor, whenever \(n\) is a power of a prime, and conjecture that this holds for all \(n\).