Fekete polynomial
In mathematics, a Fekete polynomial is a polynomial
- [math]\displaystyle{ f_p(t):=\sum_{a=0}^{p-1} \left (\frac{a}{p}\right )t^a\, }[/math]
where [math]\displaystyle{ \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-functions, and indeed to Dirichlet himself. They have acquired the name of Michael Fekete, who observed that the absence of real zeroes t of the Fekete polynomial with 0 < t < 1 implies an absence of the same kind for the L-function
- [math]\displaystyle{ 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, Zeros of Fekete polynomials, arXiv e-print math.NT/9906214, June 16, 1999.
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