Zeros of Fekete polynomials
Annales de l'Institut Fourier, Volume 50 (2000) no. 3, p. 865-889

For p an odd prime, we show that the Fekete polynomial f p (t)= a=0 p-1 a pt a has 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)+ n:n0,1 δ n /(x-n) has a zero in ]0,1[, where each δ n is ±1 with y 1/2. In fact f p (t) has absolute value p at each primitive pth root of unity, and we show that if |f p (e(2iπ(K+τ)/p))|<ϵp for some τ]0,1[ then there is a zero of f close to this arc.

Si p est un nombre premier, nous démontrons que le polynôme de Fekete f p (t)= a=0 p-1 a pt a a 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)+ n:n0,1 δ n /(x-n) a un zéro dans ]0,1[, où chaque δ n est ±1 avec probabilité 1/2. En fait la valeur absolue de f p (t) est p à chaque racine primitive p-ème de l’unité, et nous démontrons que si |f p (e(2iπ(K+τ)/p))|<ϵp avec τ]0,1[, alors il y a un zéro de f près de cet arc.

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {3},
     year = {2000},
     pages = {865-889},
     doi = {10.5802/aif.1776},
     zbl = {01478807},
     mrnumber = {2001h:11108},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_3_865_0}
}
Conrey, Brian; Granville, Andrew; Poonen, Bjorn; Soundararajan, K. Zeros of Fekete polynomials. Annales de l'Institut Fourier, Volume 50 (2000) no. 3, pp. 865-889. doi : 10.5802/aif.1776. http://www.numdam.org/item/AIF_2000__50_3_865_0/

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