Fekete polynomial

Type of polynomial
Roots of the Fekete polynomial for p = 43

In mathematics, a Fekete polynomial is a polynomial

f p ( t ) := ∑ a = 0 p − 1 ( a p ) t a {\displaystyle f_{p}(t):=\sum _{a=0}^{p-1}\left({\frac {a}{p}}\right)t^{a}\,}

where ( ⋅ p ) {\displaystyle \left({\frac {\cdot }{p}}\right)\,} 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

L ( s , x p ) . {\displaystyle L\left(s,{\dfrac {x}{p}}\right).\,}

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.
  • Brian Conrey, Andrew Granville, Bjorn Poonen and Kannan Soundararajan, Zeros of Fekete polynomials, arXiv e-print math.NT/9906214, June 16, 1999.