The distribution of zeros of zeta'(s) and gaps between zeros of zeta(s)

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.