The distribution of zeros of $\zeta'(s)$ and gaps between zeros of $\zeta(s)$

TL;DR

Under the Riemann Hypothesis, the paper establishes a precise relationship between small gaps of zeros of (s) and the distribution of zeros of its derivative, '(s), providing progress on a conjecture by Radziwi.

Contribution

The paper proves a new link between small gaps of (s) zeros and zeros of '(s), advancing understanding of their distribution under the Riemann Hypothesis.

Findings

Zeros of '(s) are located within specific regions near small gaps of (s) zeros.

Half of Radziwi's conjecture is proved in a stronger form.

The results relate zero gaps to the presence of zeros of '(s).

Abstract

Assume the Riemann Hypothesis, and let $\gamma^+>\gamma>0$ be ordinates of two consecutive zeros of $\zeta(s)$. It is shown that if $\gamma^+-\gamma < v/ \log \gamma $ with $v<c$ for some absolute positive constant $c$, then the box $$ \{s=\sigma+it: 1/2<\sigma<1/2+v^2/4\log\gamma, \gamma\le t\le \gamma^+\} $$ contains exactly one zero of $\zeta'(s)$. In particular, this allows us to prove half of a conjecture of Radziwi{\l}{\l} in a stronger form. Some related results on zeros of $\zeta(s)$ and $\zeta'(s)$ are also obtained.