Fekete polynomial References External links Navigation menuZeros of Fekete polynomials
PolynomialsZeta and L-functions
mathematicspolynomialLegendre symbolDirichlet L-functionsPeter Gustav Lejeune DirichletMichael FeketeL-functionnumber theorySiegel zero
Roots of the Fekete polynomial for p = 43
In mathematics, a Fekete polynomial is a polynomial
- fp(t):=∑a=0p−1(ap)ta{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 Peter Gustav Lejeune 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,xp).{displaystyle Lleft(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, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
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.
