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

Abstract

The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, $ξ(s)$. Thus, if the Riemann Hypothesis is true for the zeta-function, it is true for $ξ(s)$. Since $ξ(s)$ is entire, the zeros of $ξ'(s)$, its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of $ξ'(s)$ 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.