Pair correlation of the zeros of the derivative of the Riemann $ξ$-function

Details Pair correlation of the zeros of the derivative of the Riemann $ξ$-function Pair correlation of the zeros of the derivative of the Riemann $ξ$-function

2008-03-04

arxiv.org/abs/0803.0425v1

Mathematics

Number Theory

28 pages. 3 figures

Scientific paper

The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, $\xi(s)$. Thus, if the Riemann Hypothesis is true for the zeta-function, it is true for $\xi(s)$. Since $\xi(s)$ is entire, the zeros of $\xi'(s)$, its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of $\xi'(s)$ under the assumption that the Riemann Hypothesis is true. We then deduce consequences about the size of gaps between these zeros and the proportion of these zeros that are simple.

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