Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis

Emanuel Carneiro, Vorrapan Chandee, Micah B. Milinovich

Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta (s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)|…