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

This paper concerns the function S ( t ) 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 t , | S ( t ) | ≤ 0.1013 log ⁡ t . {equation*} |S(t)| ≤ 0.1013 log t. {equation*} Theorem 2 makes the above result explicit, viz. it enables one to select values of a a and b b such that, for t > t 0 t>t₀ , | S ( t ) | ≤ a + b log ⁡ t . {equation*} |S(t)| ≤ a + blog t. {equation*}