# Large Oscillations of the Argument of the Riemann Zeta-function

## TL;DR

Under the Riemann hypothesis, the paper proves that the argument of the Riemann zeta-function exhibits large oscillations of a specific order, refining previous omega results and aligning with recent findings.

## Contribution

The paper establishes a new lower bound for the oscillations of the argument of the Riemann zeta-function assuming the Riemann hypothesis, improving classical results.

## Key findings

- Proves $S(t)= ext{Omega}_ ext{pm}(rac{ ext{log} t ext{log} ext{log} ext{log} t}{ ext{log} ext{log} t})$ under RH.
- Matches recent omega results by Bondarenko and Seip.
- Refines classical oscillation bounds for $S(t)$.

## Abstract

Let $S(t)$ denote the argument of the Riemann zeta-function, defined as $$ S(t)=\dfrac{1}{\pi}\,\Im\log\zeta(1/2+it). $$ Assuming the Riemann hypothesis, we prove that $$ S(t)=\Omega_{\pm}\bigg(\dfrac{\log t\log\log\log t}{\log\log t}\bigg). $$ This improves the classical omega results of Montgomery and matches with the $\Omega$-result obtained by Bondarenko and Seip.