Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1140.11036
Zudilin, V.V.
(Zudilin, Wadim)
An essay on the irrationality measure of $\pi$ and other logarithms. (Russian)
[J] Chebyshevski\u\i\ Sb. 5, No. 2(10), 49-65 (2004). ISSN 2226-8383

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 http://arxiv.org/abs/math/0404523.
[Wadim Zudilin (Moskva)]

MSC 2000:: *11J82 Measures of irrationality and of transcendence
11J72 Irrationality
11J91 Transcendence theory of other special functions

Keywords: irrationality measure; irrationality exponent; $\pi$; logarithm; Gauss hypergeometric function; Appell hypergeometric function; transfinite diameter; rational approximation

Citations: Zbl 0635.10025; Zbl 0794.11028; Zbl 0776.11033; Zbl 0632.10034; Zbl 1208.11086; Zbl 1169.11032; Zbl 1197.11083

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]