Diophantine approximations of \(\log 2\) and other logarithms. (English. Russian original) Zbl 1201.11073

Math. Notes 83, No. 3, 389-398 (2008); translation from Mat. Zametki 83, No. 3, 428-438 (2008).
The paper deals with a new method concerning the estimation of the measure of irrationality for the numbers \(\log 2\) and \(\log\frac 53\). The author also demonstrates the estimation for approximations to the number \(\log 2\) by numbers from the field \(\mathbb Q(\sqrt 2)\). Let \(k\) be a positive integer. Then the estimation for approximations to the number \(\log\frac{\sqrt{4k^2+1}-1}{2k}\) by numbers from the field \(\mathbb Q(\sqrt{4k^2+1})\) is included. The proofs are in the spirit of Beukers’ proof of the irrationality of \(\zeta (3)\).


11J17 Approximation by numbers from a fixed field
11J82 Measures of irrationality and of transcendence


Zbl 0635.10025
Full Text: DOI


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