Geodesic
On irrationality measure $\mathrm{arctg}\, \frac {1}{3}$
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2019), pp. 69-75.

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We investigate the arithmetic properties of the value $\mathrm{arctg}\, \frac {1}{3}$. We elaborate special integral construction with the property of symmetry for evaluating irrationality measure of this number. We research linear form, generated by this integral, and prove a new result for extent of the irrationality of $\mathrm{arctg}\, \frac {1}{3}$, which improves the previous one.
Keywords: irrationality measure, linear form. 
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     title = {On irrationality measure $\mathrm{arctg}\, \frac {1}{3}$},
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V. Kh. Salikhov; M. G. Bashmakova. On irrationality measure $\mathrm{arctg}\, \frac {1}{3}$. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2019), pp. 69-75. http://geodesic.mathdoc.fr/item/IVM_2019_1_a6/

[1] Salikhov V. Kh., “O mere irratsionalnosti $\ln3$”, Dokl. RAN, 417:6 (2007), 753–755 | Zbl

[2] Wu Q., Wang L., “On the irrationality measure of $\log 3$”, J. Number Theory, 142 (2014), 264–273 | DOI | MR | Zbl

[3] Huttner M., “Irrationalité de certaines intégrales hypergéométriques”, J. Number Theory, 26 (1987), 166–178 | DOI | MR | Zbl

[4] Heimonen A., Matala-Aho T., Väänänen K., “On irrationality measures of the values of Gauss hypergeometric function”, Manuscripta Math., 81:1–2 (1993), 183–202 | DOI | MR | Zbl

[5] Heimonen A., Matala-Aho T., V\"{aä}nänen K., “An application of Jacobi type polynomials to irrationality measures”, Bull. Austral. Math. Soc., 50:2 (1994), 225–243 | DOI | MR | Zbl

[6] Tomashevskaya E. B., “O mere irratsionalnosti chisla $\ln5+ \pi/2$ i nekotorykh drugikh chisel”, Chebyshevsk. sb., 8:2 (2007), 97–108

[7] Bashmakova M. G., “O priblizhenii znachenii gipergeometricheskoi funktsii Gaussa ratsionalnymi drobyami”, Matem. zametki, 88:6 (2010), 785–797 | DOI | Zbl

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

[9] Wu Q., “On the linear independence measure of logarithms of rational numbers”, Math. Comput., 72:242 (2002), 901–911 | DOI | MR

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

[11] Marcovecchio R., “The Rhin–Viola method for $\ln 2$”, Acta Arith., 139:2 (2009), 147–184 | DOI | MR | Zbl