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The irrationality of $\log(1+1/q) \log(1-1/q)$

Author: Masayoshi Hata
Journal: Trans. Amer. Math. Soc. 350 (1998), 2311-2327
MSC (1991): Primary 11J72; Secondary 11J82
MathSciNet review: 1390038
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Abstract: We shall show that the numbers $1, \log (1+ 1/q), \log (1-1/q)$ and
$\log (1+1/q)\log (1-1/q)$ are linearly independent over $\mathbf{Q}$ for any natural number $q \ge 54$. The key is to construct explicit Padé-type approximations using Legendre-type polynomials.

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. D.V. Chudnovsky and G.V. Chudnovsky, Applications of Padé approximations to diophantine inequalities in values of G-functions, Lecture Notes in Math., vol. 1135, Springer-Verlag, 1985 pp. 9-51. MR 87a:10062
  • 3. A.I. Galochkin, Lower bounds of polynomials in the values of a certain class of analytic functions, Mat. Sbornik 95 (1974), 396-417 (Russian); English transl., Math. USSR Sb. 24 (1974), 385-407. MR 50:9806
  • 4. -, a private communication.
  • 5. M. Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99-125. MR 91i:11081
  • 6. -, On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. 69 (1990), 133-173. MR 91m:11048
  • 7. -, Rational approximations to the dilogarithm, Trans. Amer. Math. Soc. 336 (1993), 363-387. MR 93e:11088
  • 8. L. Lewin, Polylogarithms and associated functions, North-Holland, New York, 1981. MR 83b:33019
  • 9. G. Rhin and C. Viola, On the irrationality measure of $\zeta (2)$, Annales Inst. Fourier 43 (1993), 85-109. MR 94b:11065

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Additional Information

Masayoshi Hata
Affiliation: Division of Mathematics, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-01, Japan

Keywords: Logarithm, linear independence measure, Pad\'{e} approximation
Received by editor(s): April 14, 1995
Received by editor(s) in revised form: March 21, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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