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Irrationality measures of \(\log 2\) and \(\pi/\sqrt{3}\). (English) Zbl 0982.11040
Following F. Beukers’ remarkably simple proof of the irrationality of \(\zeta (3)\) [Bull. Lond. Math. Soc. 11, 268-272 (1979; Zbl 0421.10023)], K. Alladi and M. L. Robinson [J. Reine Angew. Math. 318, 137-155 (1980; Zbl 0425.10039)] discovered similar rational approximations to some infinite series and found some good irrationality measures for them. (Given a real number \(x\), we say that \(\mu \geq 2\) is an irrationality measure of \(x\) if for all \(\varepsilon>0\) there exists \(c(\varepsilon)\) for which the inequality \(|x-p/q|\geq c(\varepsilon)q^{-\mu-\varepsilon}\) is true for any integer \(p\) and any positive integer \(q\).) Let \(\mu(x)\) be the infimum over all possible \(\mu\). Their method allows \(c(\varepsilon)\) to be found effectively for numbers like \(\log 2\) and \(\pi/ \sqrt{3}= 2\sqrt{3} \arctan (1/\sqrt{3})\) and thus give estimates on \(\mu(\log 2)\) and \(\mu(\pi/\sqrt{3})\) from above. It is based on considering certain integrals involving the \(n\)-th Legendre type polynomial \(L_n(z)=\big(z^n(1-z)^n\big)^{(n)}/ n !\).
Many authors including the Chudnovskys, Rukhadze, Hata, Rhin, Viola, Huttner, Väänänen, Reyssat, the reviewer and others found some improvements of the irrationality measures mainly by replacing \(L_n(z)\) by other polynomials. For instance, Rukhadze and the reviewer in 1987 used the polynomials \(L_n(\theta,z)=\big(z^n(1-z)^n\big)^{(n-[\theta n])}/ (n-[\theta n]) !\), where \(\theta\) is some fixed positive number, to obtain the inequalities for \(\mu(\log 2)\) and \(\mu(\pi/\sqrt{3})\), respectively.
The best known estimates for irrationality measures of the numbers in the title are due to Rukhadze \(\mu(\log 2) \leq 3.892\) (her result was rediscovered several times) and Hata \(\mu(\pi/ \sqrt{3}) \leq 4.602\), respectively. Hata also found the best so far irrationality measure for \(\pi\): \(\mu(\pi) \leq 8.017\). It is sometimes not clear whether the constructions considered by different authors which look different give the same or different linear forms.
The aim of the present paper is to investigate the links between different constructions and propose some more general polynomials. The author in particular performed extensive computations which show that, unless there is a completely new idea, the above inequality for \(\mu(\log 2)\) cannot be improved just by replacing the polynomial \(L_n(\theta, z)\) by a “better” one.

MSC:
11J82 Measures of irrationality and of transcendence
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11J04 Homogeneous approximation to one number
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References:
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