$L_q$ norms and Mahler measure of Fekete polynomials

Abstract

We show that the distribution of the values of Fekete polynomials $F_p$ on the unit circle is governed, as $p\to \infty$, by an explicit limiting (non-Gaussian) random point process.This allows us to prove that the Mahler measure of $F_p$ satisfies $$M_0(F_p)\sim k_0\sqrt{p},$$ as $p\to \infty$ where $k_0=0.74083\dots ,$ thus solving an old open problem. Further, we obtain an asymptotic formula for all moments $\Vert F_p{\Vert }_q$ with $0<q<\infty ,$ resolving another open problem and improving previous results of Günther and Schmidt (who treated the case $q=2k,$ $k\in \mathbb{N}$).