Advertisement

Journal of Mathematical Sciences

, Volume 182, Issue 4, pp 539–551 | Cite as

Approximations of some logarithms by numbers from the fields \( \mathbb{Q} \) and \( \mathbb{Q}\left( {\sqrt {d} } \right) \)

  • E. S. Salnikova
Article
  • 28 Downloads

Abstract

Estimates for approximations to logarithms of rational numbers by rational numbers and quadratic irrationalities are established.

Keywords

Linear Form Rational Number Rational Approximation Hypergeometric Function Simultaneous Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. Amoroso and C. Viola, “Approximation measures for logarithms of algebraic numbers,” Ann. Scu. Norm. Sup. Pisa Cl. Sci., 30, 225–249 (2001).MathSciNetMATHGoogle Scholar
  2. 2.
    M. Hata, “Legendre type polynomials and irrationality measures,” J. Reine Angew. Math., 407, No. 1, 99–125 (1990).MathSciNetMATHGoogle Scholar
  3. 3.
    M. Hata, “Irrationality measures of the values of hypergeometric functions,” Acta Arith., 60, 335–347 (1992).MathSciNetMATHGoogle Scholar
  4. 4.
    M. Hata, “Rational approximations to π and some other numbers,” Acta Arith., 63, No. 4, 335–349 (1993).MathSciNetMATHGoogle Scholar
  5. 5.
    A. Heimonen, T. Matala-aho, and K. Väänänen, “On irrationality measures of the values of Gauss hypergeometric function,” Manuscripta Math., 81, No. 1, 183–202 (1993).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    A. Heimonen, T. Matala-aho, and K. Väänänen, “An application of Jacobi type polynomials to irrationality measures,” Bull. Austr. Math. Soc., 50, No. 2, 225–243 (1994).MATHCrossRefGoogle Scholar
  7. 7.
    M. Huttner, “Irrationalité de certaines intégrales hypergéométriques,” J. Number Theory, 26, 166–178 (1987).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    G. Rhin, “Approximants de Padé et mesures effectives d’irrationalité,” in: S´eminaire de Théorie des Nombres, Paris 1985–86, Progress Math., Vol. 71, Birkhäuser (1987), pp. 155–164.Google Scholar
  9. 9.
    E. A. Rukhadze, “Lower bounds for approximation ln 2 by rational numbers,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 25–29 (1987).Google Scholar
  10. 10.
    V. Kh. Salikhov, “On irrationality measure for log 3,” Dokl. Ross. Akad. Nauk, 417, No. 6, 753–755 (2007).MathSciNetGoogle Scholar
  11. 11.
    V. Kh. Salikhov and E. S. Salnikova, “Diophantine approximations to logarithm of the golden ratio,” Vestn. Bryansk. Gos. Tekhn. Univ., No. 1, 111–119 (2007).Google Scholar
  12. 12.
    E. S. Salnikova, “Lower bounds for approximations to log 2 by quadratic irrationalities,” Vestn. Bryansk. Gos. Tekhn. Univ., No. 2, 109–114 (2007).Google Scholar
  13. 13.
    E. S. Salnikova, “Diophantine approximations to log 2 and to other logarithms,” Mat. Zametki, 83, No. 3, 428–438 (2008).MathSciNetGoogle Scholar
  14. 14.
    E. B. Tomashevskaya, “On Diophantine approximations of values of the function log x,” Fundam. Prikl. Mat., 16, No. 6, 157–166 (2010).MathSciNetGoogle Scholar
  15. 15.
    Q. Wu, “On the linear independence measure of logarithms of rational numbers,” Math. Comp., 72, No. 242, 901–911 (2002).CrossRefGoogle Scholar
  16. 16.
    W. W. Zudilin, “A survey on irrationality measures for π and other logarithms,” Chebyshevskii Sb., 5, No. 2, 49–65 (2004).MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Bryansk State Technical UniversityBryanskRussia

Personalised recommendations