Zeros of Fekete polynomials

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

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 3, page 865-889
  • ISSN: 0373-0956

Abstract

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For p an odd prime, we show that the Fekete polynomial f p ( t ) = ∑ a = 0 p - 1 a p t 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 ∈ ℤ : n ≠ 0 , 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 p th root of unity, and we show that if | f p ( e ( 2 i π ( K + τ ) / p ) ) | < ϵ p for some τ ∈ ] 0 , 1 [ then there is a zero of f close to this arc.

How to cite

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Conrey, Brian, et al. "Zeros of Fekete polynomials." Annales de l'institut Fourier 50.3 (2000): 865-889. <http://eudml.org/doc/75441>.

@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 $0.500813&gt;\kappa _0&gt;0.500668$. 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 $\vert f_p( e(2i\pi (K+\tau )/p))\vert &lt; \epsilon \sqrt\{p\}$ 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 $0.500813&gt;\kappa _0&gt;0.500668$. 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 $\vert f_p( e(2i\pi (K+\tau )/p))\vert &lt; \epsilon \sqrt{p}$ 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 -

References

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  2. [2] H. DAVENPORT, Multiplicative Number Theory (2nd ed.), Springer-Verlag, New York, 1980. Zbl0453.10002MR82m:10001
  3. [3] P. ERDÖS and P. TURÁN, On the distribution of roots of polynomials, Ann. of Math., 51 (1950), 105-119. Zbl0036.01501MR11,431b
  4. [4] M. FEKETE and G. PÓLYA, Über ein Problem von Laguerre, Rend. Circ. Mat. Palermo, 34 (1912), 89-120. JFM43.0145.02
  5. [5] H.L. MONTGOMERY, An exponential polynomial formed with the Legendre symbol, Acta Arithmetica, 37 (1980), 375-380. Zbl0369.10024MR82a:10041
  6. [6] G. PÓLYA, Verschiedene Bemerkung zur Zahlentheorie, Jber. deutsch Math. Verein, 28 (1919), 31-40. JFM47.0882.06
  7. [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. Zbl0605.60063MR87m:42001
  8. [8] A. WEIL, Sur les fonctions algébriques à corps de constantes fini, C. R. Acad. Sci., Paris, 210 (1940), 592-594. Zbl0023.29401MR2,123dJFM66.0135.01

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