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

Abstract

We prove that for any zero ${\beta }^{\prime }+i{\gamma }^{\prime }$ of ${\zeta }^{\prime }(s)$ there exists a zero $\beta +i\gamma$ of $\zeta (s)$ such that $|\gamma -{\gamma }^{\prime }|\ll \sqrt{|{\beta }^{\prime }-\frac{1}{2}|},$ and we provide some other related results.

arXiv commentIn the revised version, by using Theorem 1 and an idea of Haseo Ki, we obtain a substantial improvement upon the older version of Theorem 3