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arXiv · 2306.07156

$L_q$ norms and Mahler measure of Fekete polynomials

Oleksiy Klurman·Youness Lamzouri·Marc Munsch
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Abstract

We show that the distribution of the values of Fekete polynomials $F_p$ on the unit circle is governed, as $p\to\infty$, by an explicit limiting (non-Gaussian) random point process.This allows us to prove that the Mahler measure of $F_p$ satisfies $$M_0(F_p)\sim k_0\sqrt{p},$$ as $p\to\infty$ where $k_0=0.74083\dots,$ thus solving an old open problem. Further, we obtain an asymptotic formula for all moments $\|F_p\|_q$ with $0<q<\infty,$ resolving another open problem and improving previous results of G\"{u}nther and Schmidt (who treated the case $q=2k,$ $k\in\mathbb{N}$).

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Oleksiy Klurman, Youness Lamzouri, Marc Munsch. 2023-06-12. $L_q$ norms and Mahler measure of Fekete polynomials. https://arxiv.org/abs/2306.07156

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