On Mahler’s Transcendence Measure for e

  • Anne-Maria Ernvall-Hytönen
  • Tapani Matala-aho
  • Louna Seppälä


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.


Diophantine approximation Hermite–Padé approximation Transcendence 

Mathematics Subject Classification

11J82 11J72 41A21 



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


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Anne-Maria Ernvall-Hytönen
    • 1
  • Tapani Matala-aho
    • 2
  • Louna Seppälä
    • 2
  1. 1.Matematik och StatistikÅbo Akademi UniversityÅboFinland
  2. 2.MatematiikkaOulun yliopistoOuluFinland

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