Negative values of the Riemann zeta function on the critical line

TL;DR

This paper proves that the Riemann zeta function attains arbitrarily large positive and negative values along the critical line, enhancing understanding of its oscillatory behavior.

Contribution

It establishes unconditionally that the zeta-function crosses the real axis infinitely often with unbounded magnitude on the critical line.

Findings

Zeta function takes arbitrarily large positive values.

Zeta function takes arbitrarily large negative values.

Confirmed oscillatory nature of the zeta function 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 unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.