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Tweaking the Beukers integrals in search of more miraculous irrationality proofs a la Apéry

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Abstract

There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero 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 \(\zeta (5)\). Inspired by Frits Beukers’ elegant rendition of Apéry’s seminal proofs of the irrationality of \(\zeta (2)\) and \(\zeta (3)\), and heavily using algorithmic proof theory, we systematically searched for other similar integrals that lead to irrationality proofs. We found quite a few candidates for such proofs, including the square-root of \(\pi \) times \(\Gamma (7/3)/\Gamma (-1/6)\) and \(\Gamma (19/6)/\Gamma (8/3)\) divided by the square-root of \(\pi \).

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ, 08854-8019, USA

    Robert Dougherty-Bliss & Doron Zeilberger

  2. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Strasse 69, 4040, Linz, Austria

    Christoph Koutschan

Authors
  1. Robert Dougherty-Bliss
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  2. Christoph Koutschan
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  3. Doron Zeilberger
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In honor of our irrational guru Wadim Zudilin, on his $$\lfloor {50\zeta (5)}\rfloor $$ ⌊ 50 ζ ( 5 ) ⌋ -th birthday.

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Dougherty-Bliss, R., Koutschan, C. & Zeilberger, D. Tweaking the Beukers integrals in search of more miraculous irrationality proofs a la Apéry. Ramanujan J 58, 973–994 (2022). https://doi.org/10.1007/s11139-021-00523-7

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  • DOI: https://doi.org/10.1007/s11139-021-00523-7

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