Distribution of the roots of the equations $Z(t)=0$, $Z'(t)=0$ in the theory of the Riemann zeta-function

Abstract

Let the symbols $\{\gamma\},\ \{t_0\};\ t_0\not=\gamma$ denote the sequences of the roots of the equations $$Z(t)=0,\quad \text{and}\qquad Z'(t)=0,$$ respectively, and $$m(t_0)=\min\{\gamma"-t_0,t_0-\gamma'\},\quad Q(t_0)=\max\{\gamma"-t_0,t_0-\gamma'\},\quad \gamma'<t_0<\gamma",$$ where $\gamma',\gamma"$ are the neighboring zeroes. We have proved the following in this paper: on the Riemann hypothesis we have $$\frac{Q(t_0)}{m(t_0)}<t_0\ln^2t_0\ln_2t_0\ln_3t_0,\quad t_0\to\infty.$$ This paper is the English version of the paper of ref. \cite{5}.