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

TL;DR

This paper investigates the distribution of roots of the Riemann zeta-function and its derivative, establishing bounds on the ratio of neighboring zero gaps under the Riemann hypothesis.

Contribution

It proves a bound on the ratio of the largest to smallest gaps between zeros of Z(t) and Z'(t) assuming the Riemann hypothesis.

Findings

Under RH, the ratio Q(t_0)/m(t_0) is bounded by t_0 ln^2 t_0 ln_2 t_0 ln_3 t_0 as t_0 approaches infinity.

The paper provides an explicit inequality relating zero gaps of Z(t) and Z'(t).

It offers insights into the spacing of zeros 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}.