The distribution of zeros of ${\zeta }^{\mathrm{\prime }}(s)$ 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$forsomeabsolutepositiveconstant$c$,thenthebox$$\{s=\sigma +it:1\mathrm{/}2<\sigma <1\mathrm{/}2+v^2\mathrm{/}4\log \gamma ,\gamma \le t\le {\gamma }^{+}\}$$containsexactlyonezeroof$\zeta’(s)$. 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 ${\zeta }^{\mathrm{\prime }}(s)$ are also obtained.