# Improved lower bounds for the Mahler measure of the Fekete polynomials

## TL;DR

This paper proves a new lower bound for the Mahler measure of Fekete polynomials, showing it grows at least proportionally to the square root of the prime, improving previous bounds.

## Contribution

It establishes a stronger universal lower bound for the Mahler measure of Fekete polynomials, advancing understanding of their size for large primes.

## Key findings

- Mahler measure of Fekete polynomials is at least c√p for some c > 1/2
- Improves previous lower bounds for large primes
- Zeros of Fekete polynomials on the unit circle are key to the analysis

## 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 $$f_p(z) := \sum_{k=1}^{p-1}{\left( \frac kp \right)z^k}\,,$$ (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 $\left(\frac 12 - \varepsilon\right)\sqrt{p}$ known before for the Mahler measure of the Fekete polynomials $f_p$ for all sufficiently large primes $p \geq c_{\varepsilon}$. Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.