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

Hata, Masayoshi

doi:10.1090/S0002-9947-98-01764-4

The irrationality of $\log (1+1/q) \log (1-1/q)$
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Trans. Amer. Math. Soc. 350 (1998), 2311-2327 Request permission

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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