Improved lower bounds for the Mahler measure of the Fekete polynomials

2017/02/20 by Tamás Erdélyi, Erdélyi, Tamás
Mathematics · #Analytic and geometric function theory #Analytic Number Theory Research #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1702.05827

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) := ∑_k=1^p-1( \frac kp )z^k , (where the coefficients are the usual Legendre symbols) is at least c√(p) for all sufficiently large primes p. This improves the lower bound (\frac 12 - ε)√(p) known before for the Mahler measure of the Fekete polynomials f_p for all sufficiently large primes p ≥ c_ε. Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

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