Supremum of the function $S_1(t)$ on short intervals

TL;DR

This paper establishes a lower bound for the maximum of the function $S_1(t)$, related to the argument of the Riemann zeta-function, on short intervals, extending previous results and providing improved Omega-results.

Contribution

It provides a new lower bound for the supremum of $S_1(t)$ on short intervals, building on and extending the work of Karatsuba and Korolev.

Findings

Derived a lower bound for $S_1(t)$ on short intervals

Extended previous supremum results to $S_1(t)$

Presented an improved Omega-result for the lower bound

Abstract

We prove a lower bound on the supremum of the function $S_1(T)$ on short intervals, defined by the integration of the argument of the Riemann zeta-function. The same type of result on the supremum of $S(T)$ have already been obtained by Karatsuba and Korolev. Our result is based on the idea of the paper of Karatsuba and Korolev. Also, we show an improved Omega-result for a lower bound.