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

Abstract

This paper concerns the function $S(T)$, where $\pi S(T)$ is the argument of the Riemann zeta-function along the critical line. The main result is that \begin{equation*} |S(T)| \leq 0.112\log T + 0.278\log \log T + 2.510, \end{equation*} which holds for all $T\geq e$.