Advertisement

Mathematical Notes

, Volume 103, Issue 3–4, pp 626–634 | Cite as

On the Irrationality Measures of Certain Numbers. II

  • A. A. PolyanskiiEmail author
Article

Abstract

For the irrationalitymeasures of the numbers \(\sqrt {2k - 1} \) arctan\(\left( {\sqrt {2k - 1} /\left( {k - 1} \right)} \right)\), where k is an even positive integer, upper bounds are presented.

Keywords

irrationality measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Hata, “Rational approximations to p and some other numbers,” Acta Arith. 63 (4), 335–349 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    V. A. Androsenko and V. Kh. Salikhov, “The Marcovecchio integral and the irrationality measure of \(\frac{\pi }{{\sqrt 3 }}\) ,” Vestnik BGTU 34 (4), 129–132 (2011).Google Scholar
  3. 3.
    R. Marcovecchio, “The Rhin–Viola method for log 2,” Acta Arith. 139 (2), 147–184 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    V. A. Androsenko, “Irrationality measure of the number \(\frac{\pi }{{\sqrt 3 }}\) ,” Izv. Ross. Akad. Nauk Ser. Mat. 79 (1), 3–20 (2015) [Izv.Math. 79 (1), 1–17 (2015)].MathSciNetCrossRefGoogle Scholar
  5. 5.
    Yu. V. Nesterenko, “On the irrationality measure of the number ln 2,” Mat. Zametki 88 (4), 549–564 (2010) [Math. Notes 88 (4), 530–543 (2010)].CrossRefzbMATHGoogle Scholar
  6. 6.
    M. G. Bashmakova, “Estimates for the exponent of irrationality for certain values of hypergeometric functions,” Mosc. J. Comb. Number Theory 1 (1), 67–78 (2011).MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. A. Polyanskii, “On the quadratic exponent of irrationality of ln 2,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 1, 25–30 (2012) [Moscow Univ.Math. Bull. 67 (1), 23–28 (2012)].MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. A. Polyanskii, “On the irrationality measure of certain numbers,” Mosc. J. Comb. Number Theory 1 (4), 80–90 (2011).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. A. Polyanskii, “Quadratic irrationality measures of certain numbers,” Vestnik Moskov. Univ. Ser. I Mat. Mekh.Moskov. Univ., No. 5, 25–29 (2013) [Moscow Univ.Math. Bull. 68 (5), 237–240 (2013)].MathSciNetzbMATHGoogle Scholar
  10. 10.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Pt. 2: Transcendental Functions (Cambridge Univ. Press, Cambridge, 1996; Editorial URSS, Moscow, 2002).zbMATHGoogle Scholar
  11. 11.
    A. A. Polyanskii, On the Irrationality Measures of Some Numbers, Cand. Sci. (Phys.–Math.) Dissertation (Izd. Moskov. Univ., Moscow, 2013), http:// polyanskii.com/ wp-content/ uploads/ thesis/ polyanskii-phd.pdf [in Russian].zbMATHGoogle Scholar
  12. 12.
    M. Hata, “ℂ2-saddle method and Beuker’s integral,” Trans. Amer.Math. Soc. 352 (10), 4557–4583 (2000).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University)Dolgoprudnyi, Moscow OblastRussia
  3. 3.Kharkevich Institute of Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

Personalised recommendations