×

Rational approximations to the dilogarithm. (English) Zbl 0768.11022

This paper continues the author’s success in breaking records for irrationality measures. Here, the dilogarithm function, \(L_ 2(z) = \sum^ \infty_{n=1} z^ n/n^ 2\), is shown to be irrational at the rational points \(z = 1/k\), where \(k\) is an integer other than \(-4,- 3,\dots,6\). The proof depends on refined Padé-type approximants and yields very good irrationality measures. These are given explicitly for the ranges \(7 \leq k \leq 18\) and \(-16 \leq k \leq -5\). For example, \[ \left| L_ 2\left({1\over 7}\right) - {p\over q}\right| \geq q^{95.0002}\quad\text{and}\quad\left| L_ 2\left({1\over 14}\right) - {p\over q}\right| \geq q^{15.275}, \] for any sufficiently large integer \(q\). The construction also leads to Padé approximants for \(\log(1-z)\) and yields measures of linear independence for \(L_ 2(1/k)\) and \(\log(1-1/k)\).

MSC:

11J82 Measures of irrationality and of transcendence
41A21 Padé approximation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] F. Beukers, A note on the irrationality of \?(2) and \?(3), Bull. London Math. Soc. 11 (1979), no. 3, 268 – 272. · Zbl 0421.10023
[2] G. V. Chudnovsky, Padé approximations to the generalized hypergeometric functions. I, J. Math. Pures Appl. (9) 58 (1979), no. 4, 445 – 476. · Zbl 0434.10023
[3] 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.
[4] 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. · Zbl 0536.10028
[5] 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. · Zbl 0755.11019
[6] A. Erdélyi et al., Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953. · Zbl 0051.30303
[7] Masayoshi Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99 – 125. · Zbl 0692.10034
[8] Masayoshi Hata, On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. (9) 69 (1990), no. 2, 133 – 173. · Zbl 0712.11040
[9] Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. · Zbl 0465.33001
[10] W. Maier, Potenzreihen irrationalen Grenzwertes, J. Reine Angew. Math. 156 (1927), 93-148. · JFM 53.0340.02
[11] Alfred van der Poorten, A proof that Euler missed…Apéry’s proof of the irrationality of \?(3), Math. Intelligencer 1 (1978/79), no. 4, 195 – 203. An informal report. · Zbl 0409.10028
[12] E. A. Rukhadze, A lower bound for the approximation of \?\?2 by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1987), 25 – 29, 97 (Russian). · Zbl 0635.10025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.