Failed attempt to disproof the Riemann Hypothesis

TL;DR

This paper reports a computer experiment calculating a specific sequence related to the Riemann Hypothesis, aiming to disprove it by detecting discrepancies, but finds no significant difference up to high precision.

Contribution

It presents a high-precision computational approach to test the Riemann Hypothesis using the Baez-Duarte criterion with two formulas for the sequence.

Findings

Discrepancy only at the 996th decimal place

High-precision calculations with thousand-digit accuracy

Potential implications for computational methods in number theory

Abstract

In this paper we are going to describe the results of the computer experiment, which in principle can rule out the Riemann Hypothesis. We use the sequence $c_k$ appearing in the \BD criterion for the RH. Namely we calculate $c_{100000}$ with thousand digits of accuracy using two different formulas for $c_k$ with the aim to disproof the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996 decimal place (accuracy of $10^{-996}$). The reported here experiment can be of interest for developers of Mathematica and PARI/GP.