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Volume 20 (2) 2014, 39-47

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

Wolf Marek

Cardinal Stefan Wyszynski University
Faculty of Mathematics and Natural Sciences. College of Sciences
ul. Wóycickiego 1/3, Auditorium Maximum, (room 113)
PL-01-938 Warsaw, Poland
e-mail: m.wolf@uksw.edu.pl

Received:

Received: 4 March 2014; revised: 9 April 2014; accepted: 11 April 2014; published online: 25 April 2014

DOI:   10.12921/cmst.2014.20.02.39-47

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 ck 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 ck with the aim to disprove the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996th decimal place (accuracy of 10−996 ). The computer experiment reported herein can be of interest for developers of Mathematica and PARI/GP.

Key words:

Baez-Duarte criterion, Riemann Hypothesis, zeta function

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