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Hypergeometric rational approximations to ζ(4)

Part of: Diophantine approximation, transcendental number theory Computational number theory

Published online by Cambridge University Press:  03 February 2020

Raffaele Marcovecchio  and
Wadim Zudilin
Raffaele Marcovecchio
Affiliation:
Dipartimento di Ingegneria e Geologia, Università di Chieti-Pescara, Viale Pindaro, 42, 65127Pescara, Italy ([email protected])
Wadim Zudilin
Affiliation:
Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500GL Nijmegen, Netherlands ([email protected])
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Abstract

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We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).

Keywords

irrationality measureπ4hypergeometric functionhypergeometric integralrational approximation

MSC classification

Primary: 11J82: Measures of irrationality and of transcendence
Secondary: 11Y60: Evaluation of constants 33C20: Generalized hypergeometric series, $_pF_q$ 33C60: Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 374 - 397
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Edinburgh Mathematical Society 2020

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