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

Author(s): Masayoshi Hata
Journal: Trans. Amer. Math. Soc. 350 (1998), 2311-2327.
MSC (1991): Primary 11J72; Secondary 11J82
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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.

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Additional Information:

Masayoshi Hata
Affiliation: Division of Mathematics, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-01, Japan
Email: hata@i.h.kyoto-u.ac.jp

DOI: 10.1090/S0002-9947-98-01764-4
PII: S 0002-9947(98)01764-4
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
Copyright of article: Copyright 1998, American Mathematical Society

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