Improved lower bounds for the Mahler measure of the Fekete polynomials

Abstract

We show that there is an absolute constant $c>1/2$ such that the Mahler measure of the Fekete polynomials $f_p$ of the form $$f_p(z):=\sum _{k=1}^{p-1}\left(\frac{k}{p}\right)z^k,$$ (where the coefficients are the usual Legendre symbols) is at least $c\sqrt{p}$ for all sufficiently large primes $p$. This improves the lower bound $\left(\frac{1}{2}-\epsilon \right)\sqrt{p}$ known before for the Mahler measure of the Fekete polynomials $f_p$ for all sufficiently large primes $p\ge c_{\epsilon }$. Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.