A quadratic integral equation

Roger D. Nussbaum

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Arithmetical functions with periodic zeros

Emil Grosswald (1975)

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On sign-changes in the remainder-term of the prime-number formula, III

Jerzy Kaczorowski (1987)

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On a diophantine equation involving quadratic characters

Karl Dilcher (1986)

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A. J. Poorten (1976)

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Zeros of Fekete polynomials

Brian Conrey, Andrew Granville, Bjorn Poonen, K. Soundararajan (2000)

Annales de l'institut Fourier

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For $p$ an odd prime, we show that the Fekete polynomial $f_{p} (t) = \sum_{a = 0}^{p - 1} (\frac{a}{p}) t^{a}$ has $\sim κ_{0} p$ zeros on the unit circle, where $0.500813 > κ_{0} > 0.500668$ . Here $κ_{0} - 1 / 2$ is the probability that the function $1 / x + 1 / (1 - x) + \sum_{n \in ℤ : n \neq 0, 1} δ_{n} / (x - n)$ has a zero in $] 0, 1 [$ , where each $δ_{n}$ is $\pm 1$ with y $1 / 2$ . In fact $f_{p} (t)$ has absolute value $\sqrt{p}$ at each primitive $p$ th root of unity, and we show that if $| f_{p} (e (2 i π (K + τ) / p)) | < ϵ \sqrt{p}$ for some $τ \in] 0, 1 [$ then there is a zero of $f$ close to this arc.

A theorem on the zeros of an entire function

Pawan Kumar Kamthan (1962-1964)

Compositio Mathematica

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