Some remarks on the Báez-Duarte criterion for the Riemann Hypothesis

Abstract

In this paper we are going to describe the results of the computer experiment, which in principle can rule out validity of the Riemann Hypothesis (RH). We use the sequence c k appearing in the Báez-Duarte criterion for the RH and compare two formulas for these numbers. We describe the mechanism of possible violation of the Riemann Hypothesis. Next we calculate c100000 with a thousand digits of accuracy using two different formulas for c k with the aim to disprove the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996 th decimal place (accuracy of 10 −996). The computer experiment reported herein can be of interest for developers of Mathematica and PARI/GP.