Zeros of Fekete polynomials

Conrey, Brian; Granville, Andrew; Poonen, Bjorn; Soundararajan, K.

Conrey, Brian ; Granville, Andrew ; Poonen, Bjorn ; Soundararajan, K.

Annales de l'Institut Fourier, Tome 50 (2000), p. 865-889 / Harvested from Numdam

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Résumé
Abstract

Si $p$ est un nombre premier, nous démontrons que le polynôme de Fekete $f_{p} (t) = \sum_{a = 0}^{p - 1} (\frac{a}{p}) t^{a}$ a $\sim κ_{0} p$ zéros sur le cercle ${| z | = 1}$ , où $0.500813 > κ_{0} > 0.500668$ . Ici $κ_{0} - 1 / 2$ est la probabilité que la fonction $1 / x + 1 / (1 - x) + \sum_{n \in ℤ : n \neq 0, 1} δ_{n} / (x - n)$ a un zéro dans $] 0, 1 [$ , où chaque $δ_{n}$ est $\pm 1$ avec probabilité $1 / 2$ . En fait la valeur absolue de $f_{p} (t)$ est $\sqrt{p}$ à chaque racine primitive $p$ -ème de l’unité, et nous démontrons que si $| f_{p} (e (2 i π (K + τ) / p)) | < ϵ \sqrt{p}$ avec $τ \in] 0, 1 [$ , alors il y a un zéro de $f$ près de cet arc.

For $p$ an odd prime, we show that the Fekete polynomial $f_{p} (t) = \sum_{a = 0}^{p - 1} (\frac{a}{p}) t^{a}$ has $\sim κ_{0} p$ zeros on the unit circle, where $0.500813 > κ_{0} > 0.500668$ . Here $κ_{0} - 1 / 2$ is the probability that the function $1 / x + 1 / (1 - x) + \sum_{n \in ℤ : n \neq 0, 1} δ_{n} / (x - n)$ has a zero in $] 0, 1 [$ , where each $δ_{n}$ is $\pm 1$ with y $1 / 2$ . In fact $f_{p} (t)$ has absolute value $\sqrt{p}$ at each primitive $p$ th root of unity, and we show that if $| f_{p} (e (2 i π (K + τ) / p)) | < ϵ \sqrt{p}$ for some $τ \in] 0, 1 [$ then there is a zero of $f$ close to this arc.

Publié le : 2000-01-01

MR 2001h:11108 | Zbl 01478807

DOI : https://doi.org/10.5802/aif.1776

@article{AIF_2000__50_3_865_0,
     author = {Conrey, Brian and Granville, Andrew and Poonen, Bjorn and Soundararajan, K.},
     title = {Zeros of Fekete polynomials},
     journal = {Annales de l'Institut Fourier},
     volume = {50},
     year = {2000},
     pages = {865-889},
     doi = {10.5802/aif.1776},
     mrnumber = {2001h:11108},
     zbl = {01478807},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2000__50_3_865_0}
}

Conrey, Brian; Granville, Andrew; Poonen, Bjorn; Soundararajan, K. Zeros of Fekete polynomials. Annales de l'Institut Fourier, Tome 50 (2000) pp. 865-889. doi : 10.5802/aif.1776. http://gdmltest.u-ga.fr/item/AIF_2000__50_3_865_0/

Bibliographie

[1] R.C. Baker and H.L. Montgomery, Oscillations of Quadratic L-functions, Analytic Number Theory (ed. B.C. Berndt et. al.), Birkhäuser, Boston (1990), 23-40. | MR 91k:11071 | Zbl 0718.11039

[2] H. Davenport, Multiplicative Number Theory (2nd ed.), Springer-Verlag, New York, 1980. | MR 82m:10001 | Zbl 0453.10002

[3] P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. of Math., 51 (1950), 105-119. | MR 11,431b | Zbl 0036.01501

[4] M. Fekete and G. Pólya, Über ein Problem von Laguerre, Rend. Circ. Mat. Palermo, 34 (1912), 89-120. | JFM 43.0145.02

[5] H.L. Montgomery, An exponential polynomial formed with the Legendre symbol, Acta Arithmetica, 37 (1980), 375-380. | MR 82a:10041 | Zbl 0439.10025

[6] G. Pólya, Verschiedene Bemerkung zur Zahlentheorie, Jber. deutsch Math. Verein, 28 (1919), 31-40. | JFM 47.0882.06

[7] M. Sambandham and V. Thangaraj, On the average number of real zeros of a random trigonometric polynomial, J. Indian Math. Soc., 47 (1983), 139-150. | MR 87m:42001 | Zbl 0605.60063

[8] A. Weil, Sur les fonctions algébriques à corps de constantes fini, C. R. Acad. Sci., Paris, 210 (1940), 592-594. | JFM 66.0135.01 | MR 2,123d | Zbl 0023.29401