The distribution of zeros of ${\zeta }^{\prime }(s)$ and gaps between zeros of $\zeta (s)$

Abstract

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 ${\zeta }^{\prime }(s)$. In particular, this allows us to prove half of a conjecture of Radziwiłł in a stronger form. Some related results on zeros of $\zeta (s)$ and ${\zeta }^{\prime }(s)$ are also obtained.

arXiv comment15 pages; first draft