Pair correlation of the zeros of the derivative of the Riemann $\xi$-function

TL;DR

This paper studies the pair correlation of zeros of the derivative of the Riemann xi-function, assuming the Riemann Hypothesis, and explores implications for zero gaps and simplicity.

Contribution

It provides the first analysis of the pair correlation of zeros of ta' under the Riemann Hypothesis, extending understanding of zero distributions.

Findings

Derived the pair correlation function for zeros of ta'

Estimated the size of gaps between zeros of ta'

Assessed the proportion of simple zeros of ta'

Abstract

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.