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A Determinantal Approach to Irrationality

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Abstract

It is a classical fact that the irrationality of a number \(\xi \in \mathbb R\) follows from the existence of a sequence \(p_n/q_n\) with integral \(p_n\) and \(q_n\) such that \(q_n\xi -p_n\ne 0\) for all n and \(q_n\xi -p_n\rightarrow 0\) as \(n\rightarrow \infty \). In this paper, we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement \(q_n\xi -p_n\rightarrow 0\) is weakened. Some applications are given, including a new proof of the irrationality of \(\pi \). Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.

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Acknowledgments

There are several inspirations for this project, the most recent one being the work [4] of Francis Brown and, in particular, his remark there: “Much more optimistically still, one might hope to prove the transcendence of \(\zeta (3)\) by optimizing our polynomial forms in \(\zeta (3)\) along the lines of [14].” The other sources of inspiration include my joint work [6] with Christian Krattenthaler, Igor Rochev, and Keijo Väänänen on (related) Hankel-determinant constructions for certain q-hypergeometric series, and also the work [10] of Hartmut Monien on (unrelated) Hankel determinants on the values of Riemann’s zeta function at positive integers. I thank all these colleagues as well as Igor Pritsker for numerous helpful chats about the topic of this project. Furthermore, I am very grateful to Stéphane Fischler, whose constructive feedback was crucial at several places of the preliminary version. Special thanks go to the anonymous referee of the journal for his healthy criticism. Part of the work was done during my visit to the Max Planck Institute for Mathematics, Bonn, in March–April 2015. I am thankful to the staff and guests of the institute for creating the unique “mathemagical” atmosphere for scientific performance.

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Authors and Affiliations

  1. School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW, 2308, Australia

    Wadim Zudilin

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  1. Wadim Zudilin
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Correspondence to Wadim Zudilin.

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Communicated by Doron S. Lubinsky.

The work is supported by Australian Research Council Grant DP170100466.

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Zudilin, W. A Determinantal Approach to Irrationality. Constr Approx 45, 301–310 (2017). https://doi.org/10.1007/s00365-016-9333-7

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  • DOI: https://doi.org/10.1007/s00365-016-9333-7

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