  Assume the Riemann Hypothesis, and let $\gamma^+>\gamma>0$ be ordinates of
two consecutive zeros of $\zeta(s)$. It is shown that if $\gamma^+-\gamma < v/
\log \gamma $ with $v<c$ for some absolute positive constant $c$, then the box
$$ \{s=\sigma+it: 1/2<\sigma<1/2+v^2/4\log\gamma, \gamma\le t\le \gamma^+\} $$
contains exactly one zero of $\zeta'(s)$. In particular, this allows us to
prove half of a conjecture of Radziwi{\l}{\l} in a stronger form. Some related
results on zeros of $\zeta(s)$ and $\zeta'(s)$ are also obtained.
