Gaps between zeros of the derivative of the Riemann \xi-function

TL;DR

Under the Riemann hypothesis, this paper studies the distribution of gaps between zeros of the derivative of the Riemann -function, establishing bounds and the existence of infinitely many small and large gaps.

Contribution

It provides new bounds on the distribution of gaps between zeros of '(s) and proves the existence of infinitely many small and large normalized gaps.

Findings

A positive proportion of gaps are less than 0.796 times the average spacing.

A positive proportion of gaps are greater than 1.18 times the average spacing.

Infinitely many normalized gaps are smaller than 0.7203 and larger than 1.5.

Abstract

Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of \xi'(s). We prove that a positive proportion of gaps are less than 0.796 times the average spacing and, in the other direction, a positive proportion of gaps are greater than 1.18 times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than 0.7203 (1.5, respectively).