$L_q$ norms and Mahler measure of Fekete polynomials
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917805366968320 |
|---|---|
| author | Klurman, Oleksiy Lamzouri, Youness Munsch, Marc |
| author_facet | Klurman, Oleksiy Lamzouri, Youness Munsch, Marc |
| contents | 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 Günther and Schmidt (who treated the case $q=2k,$ $k\in\mathbb{N}$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07156 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $L_q$ norms and Mahler measure of Fekete polynomials Klurman, Oleksiy Lamzouri, Youness Munsch, Marc Number Theory Classical Analysis and ODEs Probability 11C08, 30C10, 60G50, 11M06 (Primary) 42A05, 11L40 (Secondary) 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 Günther and Schmidt (who treated the case $q=2k,$ $k\in\mathbb{N}$). |
| title | $L_q$ norms and Mahler measure of Fekete polynomials |
| topic | Number Theory Classical Analysis and ODEs Probability 11C08, 30C10, 60G50, 11M06 (Primary) 42A05, 11L40 (Secondary) |
| url | https://arxiv.org/abs/2306.07156 |