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Zeros of Fekete polynomials

Brian Conrey, Andrew Granville, Bjorn Poonen, K. 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 ∈ ℤ : 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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