Upper and lower bounds for the function S(t) on the short intervals

TL;DR

This paper proves, assuming the Riemann Hypothesis, that the argument of the Riemann zeta function attains very large positive and negative values within extremely short intervals, highlighting extreme oscillations.

Contribution

It establishes the existence of large oscillations of the zeta function's argument on short intervals under RH, providing new bounds and insights.

Findings

Existence of large positive and negative values of S(t) on short intervals

Conditional results assuming the Riemann Hypothesis

Highlights extreme oscillations of the zeta function's argument

Abstract

We prove under RH the existence of a very large positive and negative values of the argument of the Riemann zeta function on a very short intervals.