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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 $\log 2$ [{\it 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)], $\pi$ [{\it M. Hata}, Acta Arith. 63, No. 4, 335--349 (1993; Zbl 0776.11033)], and $\log 3$ [{\it 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 $c\in\Bbb R$, for which the inequality $\vert \gamma-p/q\vert <\vert q\vert ^{-c}$ has only finitely many solutions in integers $p$ and $q\ne0$. It seems quite remarkable that the long-standing records of Hata and Rhin have been recently broken by Salikhov, who proves the estimates $\mu(\pi)<7.60630853$ [{\it 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 $\mu(\log 3)<5.125$ [{\it 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 {\it 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 \url{http://arxiv.org/abs/math/0404523}.

11J82Measures of irrationality and of transcendence
11J91Transcendence theory of other special functions