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

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 $\sqrt{\log\log\gamma'}$.