Geodesic
Diophantine Approximations of the Number~$\pi$ by Numbers from the Field $\mathbb Q(\sqrt{3})$
Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 912-922.

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We prove an estimate of the irrationality measure of any nonzero number of the form $r_1\pi+r_2\pi/\sqrt{3}$, $r_1,r_2\in\mathbb Q(\sqrt{3})$.
Mots-clés : Diophantine approximation
Keywords: the number $\pi$, the field $\mathbb Q(\sqrt{3})$, irrationality measure of a number, Kummer's formula, hypergeometric function, saddle-point method. 
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E. B. Tomashevskaya. Diophantine Approximations of the Number~$\pi$ by Numbers from the Field $\mathbb Q(\sqrt{3})$. Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 912-922. http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a9/

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

[2] F. Amoroso, C. Viola, “Approximation measures for logarithms of algebraic numbers”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30:1 (2001), 225–249 | MR | Zbl

[3] A. Heimonen, T. Matala-Aho, K. Väänänen, “An application of Jacobi type polynomials to irrationality measures”, Bull. Austral. Math. Soc., 50:2 (1994), 225–243 | DOI | MR | Zbl

[4] M. Hata, “Irrationality measures of the values of hypergeometric functions”, Acta Arith., 60:4 (1992), 335–347 | MR | Zbl

[5] V. Kh. Salikhov, “O mere irratsionalnosti $\ln 3$”, Dokl. RAN, 417:6 (2007), 753–755

[6] V. Kh. Salikhov, E. S. Salnikova, “Diofantovy priblizheniya logarifma "zolotogo secheniya”, Vestn. BGTU, 2007, no. 1, 111–119

[7] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii: Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973 | MR | Zbl