Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$\mathbf{C}^{2}$-saddle method and Beukers' integral

Author: Masayoshi Hata
Journal: Trans. Amer. Math. Soc. 352 (2000), 4557-4583
MSC (2000): Primary 11J82; Secondary 30E99
Published electronically: June 8, 2000
MathSciNet review: 1641099
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers' double integral. The two-dimensional version of the saddle method, which we call $\mathbf{C}^{2}$-saddle method, is applied.

References [Enhancements On Off] (What's this?)

  • 1. F. Beukers, A note on the irrationality of $\zeta (2)$ and $\zeta (3)$, Bull. London Math. Soc. 11 (1979) 268-272. MR 81j:10045
  • 2. H. Cohen, Accélération de la convergence de certaines récurrences linéaires, Séminaire de Théorie des Nombres, Grenoble, 1980, 47p.
  • 3. J. Dieudonné, Calcul infinitésimal, Hermann, Paris, 1968. MR 37:2557
  • 4. M. Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990) 99-125. MR 91i:11081
  • 5. M. Hata, Rational approximations to $\pi $ and some other numbers, Acta Arithmetica 63 (1993) 335-349. MR 94e:11082
  • 6. M. Hata, Rational approximations to the dilogarithm, Trans. Amer. Math. Soc. 336 (1993) 363-387. MR 93e:11088
  • 7. M. Hata, A note on Beukers' integral, J. Austral. Math. Soc. (Series A) 58 (1995) 143-153. MR 96c:11081
  • 8. L. Lewin, Polylogarithms and associated functions North-Holland, New York, 1981. MR 83b:35019
  • 9. E. Reyssat, Mesures de transcendance pour les logarithmes de nombres rationnels, Progr. Math., vol. 31, Birkhäuser, 1983 pp. 235-245. MR 85b:11060
  • 10. G. Rhin and C. Viola, On a permutation group related to $\zeta (2)$, Acta Arithmetica 77 (1996) 23-56. MR 97m:11099
  • 11. E.A. Rukhadze, A lower bound for the approximation of $\ln 2$by rational numbers, Vestnik Moskov. Univ. Ser. I Math. Mekh. no. 6 (1987) 25-29 (Russian). MR 89b:11064

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11J82, 30E99

Retrieve articles in all journals with MSC (2000): 11J82, 30E99

Additional Information

Masayoshi Hata
Affiliation: Division of Mathematics, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-8501, Japan

Keywords: Saddle method, simultaneous approximation, non-quadraticity measure
Received by editor(s): July 14, 1997
Received by editor(s) in revised form: August 26, 1998
Published electronically: June 8, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society