Zeros of Fekete polynomials

@article{Conrey2000,
abstract = {For $p$ an odd prime, we show that the Fekete polynomial $f_p(t)=\sum ^\{p-1\}_\{a=0\} \big (\{a\over p\}\big ) t^a$ 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 $1/x+1/(1-x) + \sum _\{n\in \{\Bbb Z\}:\ n\ne 0,1\} \delta _n/(x-n)$ has a zero in $]0,1[$, where each $\delta _n$ is $\pm 1$ with y $1/2$. In fact $f_p(t)$ has absolute value $\sqrt\{p\}$ at each primitive $p$th root of unity, and we show that if $\vert f_p( e(2i\pi (K+\tau )/p))\vert < \epsilon \sqrt\{p\}$ for some $\tau \in ]0,1[$ then there is a zero of $f$ close to this arc.},
author = {Conrey, Brian, Granville, Andrew, Poonen, Bjorn, Soundararajan, K.},
journal = {Annales de l'institut Fourier},
keywords = {Fekete polynomials; zeros of polynomials},
language = {eng},
number = {3},
pages = {865-889},
publisher = {Association des Annales de l'Institut Fourier},
title = {Zeros of Fekete polynomials},
url = {http://eudml.org/doc/75441},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Conrey, Brian
AU - Granville, Andrew
AU - Poonen, Bjorn
AU - Soundararajan, K.
TI - Zeros of Fekete polynomials
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 3
SP - 865
EP - 889
AB - For $p$ an odd prime, we show that the Fekete polynomial $f_p(t)=\sum ^{p-1}_{a=0} \big ({a\over p}\big ) t^a$ 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 $1/x+1/(1-x) + \sum _{n\in {\Bbb Z}:\ n\ne 0,1} \delta _n/(x-n)$ has a zero in $]0,1[$, where each $\delta _n$ is $\pm 1$ with y $1/2$. In fact $f_p(t)$ has absolute value $\sqrt{p}$ at each primitive $p$th root of unity, and we show that if $\vert f_p( e(2i\pi (K+\tau )/p))\vert < \epsilon \sqrt{p}$ for some $\tau \in ]0,1[$ then there is a zero of $f$ close to this arc.
LA - eng
KW - Fekete polynomials; zeros of polynomials
UR - http://eudml.org/doc/75441
ER -