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Transactions of the American Mathematical Society

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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.

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

Keywords: Logarithm, linear independence measure, Padé 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