Biological half-life: Difference between revisions
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[[File:Polynome de fekete 43.svg|thumbnail|300px|right|Roots of the Fekete polynomial for p = 43]] | |||
In [[mathematics]], a '''Fekete polynomial''' is a [[polynomial]] | |||
:<math>f_p(t):=\sum_{a=0}^{p-1} \left (\frac{a}{p}\right )t^a\,</math> | |||
where <math>\left(\frac{\cdot}{p}\right)\,</math> is the [[Legendre symbol]] modulo some integer ''p'' > 1. | |||
These polynomials were known in nineteenth-century studies of [[Dirichlet L-function]]s, and indeed to [[Peter Gustav Lejeune Dirichlet]] himself. They have acquired the name of [[Michael Fekete]], who observed that the absence of real zeroes ''a'' of the Fekete polynomial with 0 < ''a'' < 1 implies an absence of the same kind for the [[L-function]] | |||
:<math> L\left(s,\dfrac{x}{p}\right).\, </math> | |||
This is of considerable potential interest in [[number theory]], in connection with the hypothetical [[Siegel zero]] near ''s'' = 1. While numerical results for small cases had indicated that there were few such real zeroes, further analysis reveals that this may indeed be a 'small number' effect. | |||
==References== | |||
* [[Peter Borwein]]: ''Computational excursions in analysis and number theory.'' Springer, 2002, ISBN 0-387-95444-9, Chap.5. | |||
==External links== | |||
* [[Brian Conrey]], [[Andrew Granville]], [[Bjorn Poonen]] and [[Kannan Soundararajan]], ''[http://arxiv.org/abs/math/9906214v1 Zeros of Fekete polynomials]'', [[arXiv]] e-print math.NT/9906214, June 16, 1999. | |||
[[Category:Polynomials]] | |||
[[Category:Zeta and L-functions]] | |||
Revision as of 00:52, 20 December 2013
In mathematics, a Fekete polynomial is a polynomial
where is the Legendre symbol modulo some integer p > 1.
These polynomials were known in nineteenth-century studies of Dirichlet L-functions, and indeed to Peter Gustav Lejeune Dirichlet himself. They have acquired the name of Michael Fekete, who observed that the absence of real zeroes a of the Fekete polynomial with 0 < a < 1 implies an absence of the same kind for the L-function
This is of considerable potential interest in number theory, in connection with the hypothetical Siegel zero near s = 1. While numerical results for small cases had indicated that there were few such real zeroes, further analysis reveals that this may indeed be a 'small number' effect.
References
- Peter Borwein: Computational excursions in analysis and number theory. Springer, 2002, ISBN 0-387-95444-9, Chap.5.
External links
- Brian Conrey, Andrew Granville, Bjorn Poonen and Kannan Soundararajan, Zeros of Fekete polynomials, arXiv e-print math.NT/9906214, June 16, 1999.