An essay on the irrationality measure of \(\pi\) and other logarithms. (Russian) Zbl 1140.11036

This mini-survey summarises key ideas used in proofs of the best known (till 2007) estimates for the irrationality exponents of \(\log 2\) [E. A. Rukhadze, Mosc. Univ. Math. Bull. 42, No. 6, 30–35 (1987); translation from Vestn. Mosk. Univ., Ser. I 1987, No. 6, 25–29 (1987; Zbl 0635.10025)], \(\pi\) [M. Hata, Acta Arith. 63, No. 4, 335–349 (1993; Zbl 0776.11033)], and \(\log 3\) [G. Rhin, Théorie des nombres, Sémin. Paris 1985/86, Prog. Math. 71, 155–164 (1987; Zbl 0632.10034)].
Recall that the irrationality exponent \(\mu=\mu(\gamma)\) of a real irrational number \(\gamma\) is the infimum of quantities \(c\in\mathbb R\), for which the inequality \(| \gamma-p/q| <| q| ^{-c}\) has only finitely many solutions in integers \(p\) and \(q\neq0\).
It seems quite remarkable that the long-standing records of Hata and Rhin have been recently broken by Salikhov, who proves the estimates \(\mu(\pi)<7.60630853\) [V. Kh. Salikhov, Russ. Math. Surv. 63, No. 3, 570–572 (2008); translation from Usp. Mat. Nauk 63, No. 3, 163–164 (2008; Zbl 1208.11086)] and \(\mu(\log 3)<5.125\) [V. Kh. Salikhov, Dokl. Math. 76, No. 3, 955-957 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 417, No. 6, 753–755 (2007; Zbl 1169.11032)], while R. Marcovecchio announces a considerable improvement of Rukhadze’s estimate for \(\log 2\); his new mark is \(\mu(\log 2)<3.57455391\) [Acta Arith. 139, No. 2, 147–184 (2009; Zbl 1197.11083)].
Reviewer’s remark: The author’s translation of the article under review into English can be found at http://arxiv.org/abs/math/0404523.


11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field
11J91 Transcendence theory of other special functions
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