L_q norms and Mahler measure of Fekete polynomials

2023/06/12 by Klurman, Oleksiy, Lamzouri, Youness, Munsch, Marc · 1 citation
#11C08 #11L40 (Secondary) #11M06 (Primary) 42A05 #30C10 #60G50 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2306.07156

Abstract

We show that the distribution of the values of Fekete polynomials F_p on the unit circle is governed, as p→∞, by an explicit limiting (non-Gaussian) random point process.This allows us to prove that the Mahler measure of F_p satisfies M₀(F_p)∼ k₀√(p), as p→∞ where k₀=0.74083…, thus solving an old open problem. Further, we obtain an asymptotic formula for all moments ‖F_p‖_q with 0

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• Distribution of mixed character sums and extremal problems for Littlewood polynomials

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• Improved lower bounds for the Mahler measure of the Fekete polynomials
• Mahler's problem and Turyn polynomials
• Zeros of Fekete polynomials
• L^q norms of Fekete and related polynomials
• Mahler measure of P_d polynomials