• Source: Fekete polynomial

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.




External links


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

Kata Kunci Pencarian:

• Fekete polynomial
• Fekete
• Michael Fekete
• Gábor Szegő
• Conformal radius
• Polygonalization
• List of lemmas
• Geometric median
• List of numerical analysis topics
• Universal Taylor series