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Triple correlation of the Riemann zeros

J. Brian ConreyNina C. Snaith — 2008

Journal de Théorie des Nombres de Bordeaux

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here,...

Zeros of Fekete polynomials

Brian ConreyAndrew GranvilleBjorn PoonenK. Soundararajan — 2000

Annales de l'institut Fourier

For p an odd prime, we show that the Fekete polynomial f p ( t ) = a = 0 p - 1 a p t a has κ 0 p zeros on the unit circle, where 0 . 500813 > κ 0 > 0 . 500668 . Here κ 0 - 1 / 2 is the probability that the function 1 / x + 1 / ( 1 - x ) + n : n 0 , 1 δ n / ( x - n ) has a zero in ] 0 , 1 [ , where each δ n is ± 1 with y 1 / 2 . In fact f p ( t ) has absolute value p at each primitive p th root of unity, and we show that if | f p ( e ( 2 i π ( K + τ ) / p ) ) | < ϵ p for some τ ] 0 , 1 [ then there is a zero of f close to this arc.

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