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

TL;DR

This paper derives explicit upper bounds for the functions $S(t)$, $S_1(t)$, and the zeta function on the critical line under the Riemann Hypothesis, and applies these to bound gaps between zeros.

Contribution

It provides the first explicit bounds for these functions assuming RH, improving understanding of their behavior and zero distribution.

Findings

Explicit bounds for $S(t)$, $S_1(t)$, and $ frac{1}{2}+it$ under RH

Comparison with unconditional bounds

Conditional explicit bounds on zero gaps

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.