An improved upper bound for the argument of the Riemann zeta-function on   the critical line II
  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 $T\geq e$. 
T. S. Trudgian
math.NT