Fekete polynomial

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In mathematics, a Fekete polynomial is a polynomial

f_p(t):=\sum_{a=0}^{p-1} \left (\frac{a}{p}\right )t^a\,

where \begin{matrix}\left(\frac{.}{p}\right)\end{matrix}\, 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

\begin{matrix} L\left(s,\frac{x}{p}\right)\end{matrix}\, .

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.

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