Transactions of the American Mathematical Society

Author(s): Masayoshi Hata
Journal: Trans. Amer. Math. Soc. 352 (2000), 4557-4583.
MSC (2000): Primary 11J82; Secondary 30E99
Posted: June 8, 2000
Retrieve article in: PDF
This article is available free of charge

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.

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

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

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS