Geodesic
On the Measure of Irrationality of the Number~$\pi$
Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 583-593.

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A new estimate of the measure of irrationality of the number $\pi$ is obtained. The previous result (M. Hata, 1993) is improved by means of another integral construction.
Keywords: measure of irrationality, approximation by rational fractions, partial-fraction expansion, Leibniz formula, linear form. 
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V. Kh. Salikhov. On the Measure of Irrationality of the Number~$\pi$. Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 583-593. http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a9/

[1] K. Mahler, “On the approximation of $\pi$”, Nederl. Akad. Wetensch. Proc. Ser. A, 56:1 (1953), 30–42 | MR | Zbl

[2] M. Mignotte, “Approximations rationnelles de $\pi$ et quelques autres nombres”, Bull. Soc. Math. France Suppl. Mém., 37 (1974), 121–132 | MR | Zbl

[3] G. V. Chudnovsky, “Hermite–Padé approximations to exponential functions and elementary estimates of the measure of irrationality of $\pi$”, The Riemann Problem, Complete Integrability and Arithmetic Applications (Bures-sur-Yvette/New York, 1979/1980), Lecture Notes in Math., 925, Springer-Verlag, Berlin, 1982, 299–322 71-84 | DOI | MR | Zbl

[4] M. Hata, “Legendre type polynomials and irrationality measures”, J. Reine Angew. Math., 407:1 (1990), 99–125 | MR | Zbl

[5] M. Hata, “A lower bound for rational approximations to $\pi$”, J. Number Theory, 43:1 (1993), 51–67 | DOI | MR | Zbl

[6] M. Hata, “Rational approximations to $\pi$ and some other numbers”, Acta Arith., 63:4 (1993), 335–349 | MR | Zbl

[7] V. Kh. Salikhov, “O mere irratsionalnosti chisla $\pi$”, UMN, 63:3 (2008), 163–164 | MR | Zbl

[8] E. S. Salnikova, “Diofantovy priblizheniya $\log2$ i drugikh logarifmov”, Matem. zametki, 83:3 (2008), 428–438 | MR | Zbl