Rational approximations to the dilogarithm
Author: Masayoshi Hata
Journal: Trans. Amer. Math. Soc. 336 (1993), 363-387
MSC: Primary 11J82; Secondary 11J72
Abstract: The irrationality proof of the values of the dilogarithmic function ${L_2}(z)$ at rational points $z = 1/k$ for every integer $k \in ( - \infty , - 5] \cup [7,\infty )$ is given. To show this we develop the method of Padé-type approximations using Legendre-type polynomials, which also derives good irrationality measures of ${L_2}(1/k)$. Moreover, the linear independence over ${\mathbf {Q}}$ of the numbers $1$, $\log (1 - 1/k)$, and ${L_2}(1/k)$ is also obtained for each integer $k \in ( - \infty , - 5] \cup [7,\infty )$ .
F. Beukers, A note on the irrationality of $\zeta (2)$ and $\zeta (3)$, Bull. London Math. Soc. 11 (1979), no. 3, 268–272. MR 554391, DOI https://doi.org/10.1112/blms/11.3.268
G. V. Chudnovsky, Padé approximations to the generalized hypergeometric functions. I, J. Math. Pures Appl. (9) 58 (1979), no. 4, 445–476. MR 566655
G. V. Chudnovsky, Measures of irrationality, transcendence and algebraic independence. Recent progress, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 11–82. MR 697257
D. V. Chudnovsky and G. V. Chudnovsky, Padé and rational approximations to systems of functions and their arithmetic applications, Number theory (New York, 1982) Lecture Notes in Math., vol. 1052, Springer, Berlin, 1984, pp. 37–84. MR 750662, DOI https://doi.org/10.1007/BFb0071540
R. Dvornicich and C. Viola, Some remarks on Beukers' integrals, Number theory, Vol. II (Budapest, 1987) Colloq. Math. Soc. János Bolyai, vol. 51, North-Holland, Amsterdam, 1990, pp. 637–657. MR 1058238
A. Erdélyi et al., Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953.
Masayoshi Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99–125. MR 1048530, DOI https://doi.org/10.1515/crll.1990.407.99
Masayoshi Hata, On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. (9) 69 (1990), no. 2, 133–173. MR 1067449
Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
W. Maier, Potenzreihen irrationalen Grenzwertes, J. Reine Angew. Math. 156 (1927), 93-148.
Alfred van der Poorten, A proof that Euler missed$\ldots $Apéry's proof of the irrationality of $\zeta (3)$, Math. Intelligencer 1 (1978/79), no. 4, 195–203. An informal report. MR 547748, DOI https://doi.org/10.1007/BF03028234
E. A. Rukhadze, A lower bound for the approximation of ${\rm ln}\,2$ by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1987), 25–29, 97 (Russian). MR 922879
F. Beukers, A note on the irrationality of $\zeta (2)$ and $\zeta (3)$, Bull. London Math. Soc. 11 (1979), 268-272. G. V. Chudnovsky, Padé approximations to the generalized hypergeometric functions. I, J. Math. Pures Appl. 58 (1979), 445-476. ---, Measures of irrationality, transcendence and algebraic independence. Recent progress (Number Theory Days, 1980), London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 11-82. D. V. Chudnovsky and G. V. Chudnovsky, Padé and rational approximations to systems of functions and their arithmetic applications, Lecture Notes in Math., vol. 1052, Springer-Verlag, 1984, pp. 37-84. R. Dvornicich and C. Viola, Some remarks on Beukers' integrals (Number Theory, Budapest), Colloq. Math. Soc. János Bolyai, vol. 51, North-Holland, 1987, pp. 637-657. A. Erdélyi et al., Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953. M. Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99-125. ---, On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. 69 (1990), 133-173. L. Lewin, Polylogarithms and associated functions, North-Holland, New York, 1981. W. Maier, Potenzreihen irrationalen Grenzwertes, J. Reine Angew. Math. 156 (1927), 93-148. A. van der Poorten, A proof that Euler missed—Apéry's proof of the irrationality of $\zeta (3)$, Math. Intelligencer 1 (1979), 195-203. E. A. Rukhadze, A lower bound for the approximation of $\ln 2$ by rational numbers, Vestnik Moskov. Univ. Ser. I Math. Mekh. (1987), no. 6, 25-29. (Russian)
Retrieve articles in Transactions of the American Mathematical Society with MSC: 11J82, 11J72
Retrieve articles in all journals with MSC: 11J82, 11J72
Keywords: Dilogarithm, irrationality measure, Padé approximation