On explicit estimates for $S(t)$, $S_1(t)$, and $\zeta(1/2+\mathrm{i}t)$ under the Riemann Hypothesis

Abstract

Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of $S(t)$, $S_1(t)$, and $\zeta\left(1/2+\mathrm{i}t\right)$ while comparing them with recently proven unconditional ones. As a corollary we obtain a conditional explicit bound on gaps between consecutive zeros of the Riemann zeta-function.