On the value-distribution of the Riemann zeta-function on the critical line

TL;DR

This paper explores the behavior of the Riemann zeta-function on the critical line, establishing mean-value results under the Riemann hypothesis and demonstrating unconditionally that it attains arbitrarily large real values.

Contribution

It provides new insights into the value distribution of the zeta-function, including mean-value existence under RH and unbounded real values unconditionally.

Findings

Mean-value of real values exists if RH is true and equals 1.

Zeta-function takes arbitrarily large real values unconditionally.

Intersections with the real axis are analyzed on the critical line.

Abstract

We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show that if the Riemann hypothesis is true, the mean-value of those real values exists and is equal to 1. Moreover, we show unconditionally that the zeta-function takes arbitrarily large real values on the critical line.