Fekete polynomial

In mathematics, a Fekete polynomial is a polynomial

:$f\_p(t):=sum\_\{a=0\}^\{p-1\}\; left\; (frac\{a\}\{p\}\; ight\; )t^a,$

where is the Legendre symbol modulo some integer "p" > 1, and the summation is for 1 ≤ "n" < "p".

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 "a" of the Fekete polynomial with 0 < "a" < 1 implies an absence of the same kind for the L-function

:$Lleft(s,dfrac\{x\}\{p\}\; ight).,$

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-Verlag, 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.