Improved Lower Bound 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 $$\begin{aligned} f_p(z) := \sum _{k=1}^{p-1}{\left( \frac{k}{p} \right) z^k} \end{aligned}$$ (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 $$\left( \frac{1}{2} - \varepsilon \right) \sqrt{p}$$ known before for the Mahler measure of the Fekete polynomials $$f_p$$ for all sufficiently large primes $$p \ge c_{\varepsilon }$$ . Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.