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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-98-01764-4
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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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

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