Tweaking the Beukers Integrals In Search of More Miraculous Irrationality Proofs À La Apéry

By Robert Dougherty-Bliss, Christoph Koutschan, and Doron Zeilberger

.pdf    .tex    Appendix(.pdf)

Appeared in the Ramanujan Journal v. 58 (2022), pages 973-994.

Posted: Jan. 13, 2021

Last update of this web-page: June 13, 2022 (to link to a brilliant comment of Jean-Paul Allouche)

In honor of Wadim Zudilin, on the occaison of his trunc(50*Zeta(5))-th birthday

There are only aleph0 rational numbers, while there are 2aleph0 real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or ζ(5).

Inspired by Frits Beukers' elegant rendition of Apéry's seminal proofs of the irrationality of ζ(2) and ζ(3), and heavily using Wilf-Zeilberger algorithmic proof theory and Koutschan's efficient Holnomic Functions programs, we systematically searched for other similar integrals, that lead to irrationality proofs. We found quite a few candidates for such proofs, including π1/2Γ(7/3)/Γ(-1/6) and π-1/2Γ(19/6)/Γ(8/3)

Added Aug. 24, 2021: The "birthday boy", Wadim Zudlin, met the challenges mentioned at the end of our paper. See here. Congratulations!

Added June 13, 2022: Jean-Paul Allouche, unaware of Wadim Zudilin's above-mentioned note, independently made the following brilliant comment.

Maple packages

Sample Input and Output for GenBeukersLog.txt

Sample Input and Output for GenBeukersZeta2.txt

Sample Input and Output for GenBeukersZeta3.txt

Articles of Doron Zeilberger

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Robert Dougherty-Bliss 's Home Page

Christoph Koutschan's Home Page