Zeros of Fekete polynomials

Si $p$ est un nombre premier, nous démontrons que le polynôme de Fekete $f_p\left(t\right)={\sum }_{a=0}^{p-1}\left(\frac{a}{p}\right)t^a$ a $\sim {\kappa }_{0}p$ zéros sur le cercle $\left\{\right|z|=1\}$, où $0.500813>{\kappa }_0>0.500668$. Ici ${\kappa }_0-1/2$ est la probabilité que la fonction $1/x+1/(1-x)+{\sum }_{n\in \mathbb{Z}:n\ne 0,1}{\delta }_n/(x-n)$ a un zéro dans $]0,1[$, où chaque ${\delta }_n$ est $\pm 1$ avec probabilité $1/2$. En fait la valeur absolue de $f_p\left(t\right)$ est $\sqrt{p}$ à chaque racine primitive $p$-ème de l’unité, et nous démontrons que si $|f_p\left(e\right(2i\pi (K+\tau )/p\left)\right)|<ϵ\sqrt{p}$ avec $\tau \in ]0,1[$, alors il y a un zéro de $f$ près de cet arc.

For $p$ an odd prime, we show that the Fekete polynomial $f_p\left(t\right)={\sum }_{a=0}^{p-1}\left(\frac{a}{p}\right)t^a$ has $\sim {\kappa }_{0}p$ zeros on the unit circle, where $0.500813>{\kappa }_0>0.500668$. Here ${\kappa }_0-1/2$ is the probability that the function $1/x+1/(1-x)+{\sum }_{n\in \mathbb{Z}:n\ne 0,1}{\delta }_n/(x-n)$ has a zero in $]0,1[$, where each ${\delta }_n$ is $\pm 1$ with y $1/2$. In fact $f_p\left(t\right)$ has absolute value $\sqrt{p}$ at each primitive $p$th root of unity, and we show that if $|f_p\left(e\right(2i\pi (K+\tau )/p\left)\right)|<ϵ\sqrt{p}$ for some $\tau \in ]0,1[$ then there is a zero of $f$ close to this arc.

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