Advertisement

On Mahler’s Transcendence Measure for e

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

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.

Keywords

Diophantine approximation Hermite–Padé approximation Transcendence 

Mathematics Subject Classification

11J82 11J72 41A21 

Notes

Acknowledgements

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

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. In: National Bureau Of Standards Applied Mathematics Series, vol. 55. Washington, DC (1964)Google Scholar
  2. 2.
    Baker, A.: Transcendental Number Theory. Cambridge University Press, Cambridge (1975)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borel, É.: Sur la nature arithmétique du nombre \(e\). C. R. Acad. Sci. Paris 128, 596–599 (1899)zbMATHGoogle Scholar
  4. 4.
    Fel’dman, N.I., Nesterenko, Y.V.: Transcendental numbers, number theory IV, 1–345. In: Encyclopaedia of Mathematical Sciences, vol. 44. Springer, Berlin (1998)Google Scholar
  5. 5.
    Hančl, J., Leinonen, M., Leppälä, K., Matala-aho, T.: Explicit irrationality measures for continued fractions. J. Number Theory 132, 1758–1769 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hata, M.: Remarks on Mahler’s transcendence measure for \(e\). J. Number Theory 54, 81–92 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hermite, C.: Sur la fonction exponentielle, C. R. Acad. Sci. 77:18–24, 74–79, 226–233, 285–293 (1873)Google Scholar
  8. 8.
    Khassa, D.S., Srinivasan, S.: A transcendence measure for \(e\). J. Indian Math. Soc. 56, 145–152 (1991)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Leppälä, K., Matala-aho, T., Törmä, T.: Rational approximations of the exponential function at rational points. J. Number Theory 179, 220–239 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mahler, K.: Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I. J. Reine Angew. Math. 166, 118–136 (1931)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Mahler, K.: Lectures on Transcendental Numbers. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  12. 12.
    Popken, J.: Sur la nature arithmétique du nombre \(e\). C. R. Acad. Sci. Paris 186, 1505–1507 (1928)zbMATHGoogle Scholar
  13. 13.
    Popken, J.: Zur Transzendenz von \(e\). Math. Z. 29, 525–541 (1929)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    SageMath, the Sage Mathematics Software System (Version 6.5): The Sage Developers (2015). http://www.sagemath.org
  15. 15.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. https://oeis.org/. Accessed 4 Jan 2017
  16. 16.
    Waldschmidt, M.: Introduction to Diophantine methods: irrationality and transcendence (2014). https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/IntroductionDiophantineMethods.pdf. Accessed 11 Jan 2017

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

Personalised recommendations