The explicit upper bound of the multiple integral of $S(t)$ on the Riemann Hypothesis

TL;DR

This paper establishes explicit upper bounds for the iterated integrals of the argument of the Riemann zeta-function, generalizing previous bounds for the first integral and providing new bounds for higher-order integrals.

Contribution

The paper extends Fujii's explicit bounds for $S(T)$ and $S_1(T)$ to higher-order integrals $S_m(T)$, offering a broader understanding of the function's behavior under the Riemann Hypothesis.

Findings

Explicit upper bounds for $S_m(T)$ are derived.

The bounds generalize previous results for $S(T)$ and $S_1(T)$.

Results contribute to the understanding of the argument of the Riemann zeta-function.

Abstract

We prove explicit upper bounds of the function $S_m(T)$, defined by the repeated integration of the argument of the Riemann zeta-function. The explicit upper bound of $S(T)$ and $S_1(T)$ have already been obtained by A. Fujii. Our result is a generalization of Fujii's results.