Conditional estimates on small distances between ordinates of zeros of $\zeta(s)$ and $\zeta'(s)$

TL;DR

This paper improves bounds on the small distances between zeros of the Riemann zeta function and its derivative, assuming the Riemann Hypothesis, by refining previous estimates with a logarithmic factor.

Contribution

It provides a sharper bound on the proximity of zeros of ta(s) and ta'(s) under RH, enhancing previous results by Garaev and Y
31ld
31r
31m.

Findings

Improved bound on zero distances assuming RH

Reduction of the previous estimate by a (log log
31
31
31
31m) factor

Strengthens understanding of zero distribution of ta(s) and ta'(s)

Abstract

Let $\beta'+i\gamma'$ be a zero of $\zeta'(s)$. In \cite{GYi} Garaev and Y{\i}ld{\i}r{\i}m proved that there is a zero $\beta+i\gamma$ of $\zeta(s)$ with $ \gamma'-\gamma \ll \sqrt{|\beta'-1/2|} $. Assuming RH, we improve this bound by saving a factor $\sqrt{\log\log\gamma'}$.