Conditional estimates on small distances between ordinates of zeros of $\zeta (s)$ and ${\zeta }^{\mathrm{\prime }}(s)$

Let ${\beta }^{\mathrm{\prime }}+i{\gamma }^{\mathrm{\prime }}$ be a zero of ${\zeta }^{\mathrm{\prime }}(s)$. 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 ${\gamma }^{\mathrm{\prime }}-\gamma \ll \sqrt{\mathrm{\mid }{\beta }^{\mathrm{\prime }}-1\mathrm{/}2\mathrm{\mid }}$. Assuming RH, we improve this bound by saving a factor $\sqrt{\log \log {\gamma }^{\mathrm{\prime }}}$.