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 2^aleph₀ 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 π^1/2Γ(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

• GenBeukersLog.txt, a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of the Alladi-Robinson integrals that occurred in their elegant treatment of irrationality measures of log(2) and many other logs.

• GenBeukersZeta2.txt, a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of Beukers' integral that occurred in his elegant rendition of Apery's miraculous proofs that Zeta(2) is irrational.

• GenBeukersZeta3.txt, a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of Beukers' integral that occurred in his elegant rendition of Apery's miraculous proofs that Zeta(3) is irrational.

Sample Input and Output for GenBeukersLog.txt

• If you want to see many candidates for Apéry-style proofs of generalized Alladi-Robinson integrals, many of them identifified (and since there either logs of rational numbers or explicit algebraic numbers will not get us famous) but many of them not yet identified, so potentially not yet proved to be rational
  The input file yields the output file

• If you want to see numerous sketches (modulo some filling-in-details for the divisibility lemmas) of Apéry-style proofs similar to the Alladi-Robinson irrationality proofs of log(c) for many rationals c, including some that may be not yet proved to be irrational (at least Maple can't identify them)
  The input file yields the output file

• If you want to see many candidates for Apéry-style proofs of super-generalized Alladi-Robinson integrals, (i.e. the 4-paramter family of constants

  int(x^a*(1-x)^b)/(1+c*x)^(d+1),x=0..1)/int(x^a*(1-x)^b)/(1+c*x)^d,x=0..1)

  many of them identified (and since there either logs of rational numbers or explicit algebraic numbers will not get us famous) but many of them not yet identified, so potentially not yet proved to be rational
  The input file yields the output file

Sample Input and Output for GenBeukersZeta2.txt

• If you want to see numerous sketches (modulo some filling-in-details for the divisibility lemmas) of Apéry-style proofs similar to Beukers' irrationality proof of Zeta(2), most of them for constants that are probably already proved to be irrational (e.g. log(2), sqrt(2), Pi*sqrt(3)) but a few that Maple can't identify, and are potentially the first proof of their irrationality
  The input file yields the output file

• If you want to see synopsises of irrationality proofs of some constants, some identified, some not yet (hence giving hope that they are not yet proved to be irrational)
  The input file yields the output file

• If you want to see lots of challenges for Wadim Zudilin, giving conjectured 3F2 evaluations. [Not related to irrationality]
  The input file yields the output file

Sample Input and Output for GenBeukersZeta3.txt

• If you want to see numerous sketches (modulo some filling-in-details for the divisibility lemmas) of Apéry-style proofs similar to Beukers' irrationality proof of Zeta(3), most of them for constants that are probably already proved to be irrational (e.g. log(2), sqrt(2), Pi*sqrt(3)) but a few that Maple can't identify, and are potentially the first proof of their irrationality
  The input file yields the output file

Doron Zeilberger's Home Page

Robert Dougherty-Bliss 's Home Page

Christoph Koutschan's Home Page