An improved upper bound for the argument of the Riemann zeta-function on the critical line II

TL;DR

This paper presents an improved upper bound for the argument of the Riemann zeta-function on the critical line, refining previous estimates and providing a tighter bound valid for all sufficiently large T.

Contribution

It introduces a sharper upper bound for |S(T)|, enhancing the understanding of the argument of the Riemann zeta-function on the critical line.

Findings

|S(T)|  0.111  \, ext{log} \, T + 0.275 \, ext{log} \, ext{log} \, T + 2.450

Bound is valid for all T  e

Improves previous estimates for the argument of the zeta-function

Abstract

This paper improves the bound on $|S(T)|$. The main result is to show that $|S(T)|\leq 0.111\log T + 0.275\log\log T + 2.450$, which is valid for all $T\geq e$.