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On Mahler’s Transcendence Measure for e

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Abstract

We present a completely explicit transcendence measure for e. This is a continuation and an improvement to the works of Borel, Mahler, and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of e. The results are based on Hermite–Padé approximations and on careful analysis of common factors in the footsteps of Hata.

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Acknowledgements

We are indebted to the anonymous referees for their critical reading and helpful suggestions.

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Authors and Affiliations

  1. Matematik och Statistik, Åbo Akademi University, Domkyrkotorget 1, 20500, Åbo, Finland

    Anne-Maria Ernvall-Hytönen

  2. Matematiikka, Oulun yliopisto, PL 8000, 90014, Oulu, Finland

    Tapani Matala-aho & Louna Seppälä

Authors
  1. Anne-Maria Ernvall-Hytönen
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  2. Tapani Matala-aho
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  3. Louna Seppälä
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Corresponding author

Correspondence to Tapani Matala-aho.

Additional information

Communicated by Edward B. Saff.

The work of the author Louna Seppälä was supported by the Magnus Ehrnrooth Foundation. The work of the author Anne-Maria Ernvall-Hytönen was supported by the Academy of Finland Project 303820 and by the Finnish Cultural Foundation.

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Ernvall-Hytönen, AM., Matala-aho, T. & Seppälä, L. On Mahler’s Transcendence Measure for e. Constr Approx 49, 405–444 (2019). https://doi.org/10.1007/s00365-018-9429-3

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  • DOI: https://doi.org/10.1007/s00365-018-9429-3

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