$\mathbf {C}^{2}$-saddle method and Beukersâ integral
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Abstract:
We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukersâ double integral. The two-dimensional version of the saddle method, which we call $\mathbf {C}^{2}$-saddle method, is applied.References
- F. Beukers, A note on the irrationality of $\zeta (2)$ and $\zeta (3)$, Bull. London Math. Soc. 11 (1979), no. 3, 268â272. MR 554391, DOI 10.1112/blms/11.3.268
- H. Cohen, Accélération de la convergence de certaines récurrences linéaires, Séminaire de Théorie des Nombres, Grenoble, 1980, 47p.
- Jean Dieudonné, Calcul infinitésimal, Hermann, Paris, 1968 (French). MR 0226971
- Masayoshi Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99â125. MR 1048530, DOI 10.1515/crll.1990.407.99
- Masayoshi Hata, Rational approximations to $\pi$ and some other numbers, Acta Arith. 63 (1993), no. 4, 335â349. MR 1218461, DOI 10.4064/aa-63-4-335-349
- Masayoshi Hata, Rational approximations to the dilogarithm, Trans. Amer. Math. Soc. 336 (1993), no. 1, 363â387. MR 1147401, DOI 10.1090/S0002-9947-1993-1147401-5
- Masayoshi Hata, A note on Beukersâ integral, J. Austral. Math. Soc. Ser. A 58 (1995), no. 2, 143â153. MR 1323987, DOI 10.1017/S1446788700038192
- R. Ya. DoktorskiÄ, On the proximity between solutions of differential equations in âlargeâ domains and in the whole space, Funktsional. Anal. i Prilozhen. 15 (1981), no. 3, 87â88 (Russian). MR 630345
- Ă. Reyssat, Mesures de transcendance pour les logarithmes de nombres rationnels, Diophantine approximations and transcendental numbers (Luminy, 1982) Progr. Math., vol. 31, BirkhĂ€user Boston, Boston, MA, 1983, pp. 235â245 (French). MR 702201, DOI 10.1007/BF02591752
- Georges Rhin and Carlo Viola, On a permutation group related to $\zeta (2)$, Acta Arith. 77 (1996), no. 1, 23â56. MR 1404975, DOI 10.4064/aa-77-1-23-56
- E. A. Rukhadze, A lower bound for the approximation of $\textrm {ln}\,2$ by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1987), 25â29, 97 (Russian). MR 922879
Additional Information
- Masayoshi Hata
- Affiliation: Division of Mathematics, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-8501, Japan
- Email: hata@i.h.kyoto-u.ac.jp
- Received by editor(s): July 14, 1997
- Received by editor(s) in revised form: August 26, 1998
- Published electronically: June 8, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 4557-4583
- MSC (2000): Primary 11J82; Secondary 30E99
- DOI: https://doi.org/10.1090/S0002-9947-00-02432-6
- MathSciNet review: 1641099