Abstract:
A new efficient construction of Diophantine approximations to Catalan's constant is presented that is based on the direct analysis of the representation of a hypergeometric function with specially chosen half-integer parameters as a series and as a double Euler integral over the unit cube. This allows one to significantly simplify the proofs of Diophantine results available in this domain and substantially extend the capabilities of the method. The sequences of constructed rational approximations are not good enough to prove irrationality, but the results established allow one to compare the quality of various constructions.
Citation:
Yu. V. Nesterenko, “On Catalan's constant”, Algebra, geometry, and number theory, Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 292, MAIK Nauka/Interperiodica, Moscow, 2016, 159–176; Proc. Steklov Inst. Math., 292 (2016), 153–170
\Bibitem{Nes16}
\by Yu.~V.~Nesterenko
\paper On Catalan's constant
\inbook Algebra, geometry, and number theory
\bookinfo Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2016
\vol 292
\pages 159--176
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3695}
\crossref{https://doi.org/10.1134/S0371968516010106}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3628459}
\elib{https://elibrary.ru/item.asp?id=25772718}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2016
\vol 292
\pages 153--170
\crossref{https://doi.org/10.1134/S0081543816010107}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000376271200010}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84971367773}
Linking options:
https://www.mathnet.ru/eng/tm3695
https://doi.org/10.1134/S0371968516010106
https://www.mathnet.ru/eng/tm/v292/p159
This publication is cited in the following 3 articles:
Nadav Ben David, Guy Nimri, Uri Mendlovic, Yahel Manor, Carlos De la Cruz Mengual, Ido Kaminer, “On the Connection Between Irrationality Measures and Polynomial Continued Fractions”, Arnold Math J., 2024
Dougherty-Bliss R., Koutschan Ch., Zeilberger D., “Tweaking the Beukers Integrals in Search of More Miraculous Irrationality Proofs a La Apery”, Ramanujan J., 58:3 (2022), 973–994
G. Raayoni, Sh. Gottlieb, Ya. Manor, G. Pisha, Y. Harris, U. Mendlovic, D. Haviv, Ya. Hadad, I. Kaminer, “Generating conjectures on fundamental constants with the Ramanujan machine”, Nature, 590:7844 (2021), 67+
Citation in format AMSBIB
Contact us: