On Balazard, Saias, and Yor's equivalence to the Riemann Hypothesis

TL;DR

This paper explores the equivalence of the Riemann Hypothesis to a specific integral condition, investigates the convergence rate of a truncated integral assuming RH, and provides a new proof of a classical Omega theorem related to the zeta-function.

Contribution

It offers new insights into the rate of convergence of a truncated integral linked to the Riemann Hypothesis and presents a novel proof of a classical Omega theorem.

Findings

Disproved a conjecture by Borwein, Bradley, and Crandall regarding the integral's behavior.

Established the rate at which the truncated integral tends to zero under RH.

Provided a new proof of a classical Omega theorem for the function S(t).

Abstract

Balazard, Saias, and Yor proved that the Riemann Hypothesis is equivalent to a certain weighted integral of the logarithm of the Riemann zeta-function along the critical line equaling zero. Assuming the Riemann Hypothesis, we investigate the rate at which a truncated version of this integral tends to zero, answering a question of Borwein, Bradley, and Crandall and disproving a conjecture of the same authors. A simple modification of our techniques gives a new proof of a classical Omega theorem for the function S(t) in the theory of the Riemann zeta-function.