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

Abstract:

This paper concerns the function $S(t)$, the argument of the Riemann zeta-function along the critical line. Improving on the method of Backlund, and taking into account the refinements of Rosser and McCurley it is proved that for sufficiently large $t$, \begin{equation*} |S(t)| \leq 0.1013 \log t. \end{equation*} Theorem 2 makes the above result explicit, viz. it enables one to select values of $a$ and $b$ such that, for $t>t_{0}$, \begin{equation*} |S(t)| \leq a + b\log t. \end{equation*}