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

Abstract

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

to appear in JNT; may differ slightly from the final version