Russian Mathematics

, Volume 63, Issue 1, pp 61–66 | Cite as

On Irrationality Measure of arctan \(\frac{1}{3}\)

  • V. Kh. SalikhovEmail author
  • M. G. BashmakovaEmail author


We investigate the arithmetic properties of the value arctan \(\frac{1}{3}\). We elaborate special integral construction with the property of symmetry for evaluating irrationality measure of this number. We research linear form, generated by this integral, and prove a new result for extent of the irrationality of arctan \(\frac{1}{3}\), which improves the previous one.

Key words

irrationality measure linear form 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Supported by Russian Found of Fundamental Research, grant No. ‘18-01-00296A.


  1. 1.
    Salikhov, V. Kh. “On Irrationality Measure of ln 3”, Dokl. RAN 417, No. 6, 753–755 (2007) [in Russian].MathSciNetzbMATHGoogle Scholar
  2. 2.
    Wu, Q., Wang, L. “On the Irrationality Measure of log 3”, J. Number Theory 142, 264–273 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Huttner, M. “Irrationalité de Certaines Intégrales Hypergéométriques”, J. Number Theory 26, 166–178 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Heimonen, A., Matala-Aho, T., Väänänen, K. “On Irrationality Measures of the Values of Gauss Hypergeometric Function”, Manuscripta Math. 81, No. 1–2, 183–202 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Heimonen, A., Matala-Aho, T., Väänänen, K. “An Application of Jacobi Type Polynomials to Irrationality Measures”, Bull. Austral. Math. Soc. 50, No. 2, 225–243 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Tomashevskaya, E. B. “On Irrationality Measures of Number ln 5 + π/2 and Some Other Numbers”, Chebyshevsk. sb. 8, No. 2, 97–108 (2007) [in Russian].MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bashmakova, M. G. textquotedblleft On Approaching of Values of Hypergeometric Gauss FunctionWith Rational Fractions”, Matem. zametki 88, No. 6, 785–797 (2010) [in Russian].CrossRefGoogle Scholar
  8. 8.
    Hata, M. “Rational Approximation to π and Some Other Numbers”, Acta Arith. 63, No. 4, 335–349 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wu, Q. “On the Linear Independence Measure of Logarithms of Rational Numbers”, Math. Comput. 72 (242), 901–911 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Salikhov, V. Kh. “On Irrationality Measures of Number π”, Usp. Mat. Nauk 63, No. 3, 163–164 (2008) [in Russian].CrossRefGoogle Scholar
  11. 11.
    Marcovecchio R. “The Rhin–Viola Method for ln 2”, Acta Arith. 139, No. 2, 147–184 (2009).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Bryansk State Technical UniversityBryanskRussia

Personalised recommendations