Geodesic
On irrationality measure of some values of $\arctan \frac{1}{n}$
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2020), pp. 32-40.

Voir la notice de l'article provenant de la source Math-Net.Ru

The aim of research is to obtain new estimates of extent of irrationality for a values $\arctan \frac{1}{5}, \arctan\frac{1}{3}.$ In this article we constructed new integral for getting irrationality measure of $\arctan \frac{1}{5}$, based on idea from work of K. Wu, 2002. We investigated linear form, generated by this integral and we found that it allow to get better estimate for this value. By the same method we constructed integral for obtaining an estimate of irrationality measure for $\arctan \frac{1}{3},$ and we also got new result for this value.
Keywords: irrationality measure, linear form. 
@article{IVM_2020_12_a3,
     author = {V. Kh. Salikhov and M. G. Bashmakova},
     title = {On irrationality measure of some values of $\arctan \frac{1}{n}$},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {32--40},
     publisher = {mathdoc},
     number = {12},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2020_12_a3/}
}
TY  - JOUR
AU  - V. Kh. Salikhov
AU  - M. G. Bashmakova
TI  - On irrationality measure of some values of $\arctan \frac{1}{n}$
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2020
SP  - 32
EP  - 40
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2020_12_a3/
LA  - ru
ID  - IVM_2020_12_a3
ER  - 
%0 Journal Article
%A V. Kh. Salikhov
%A M. G. Bashmakova
%T On irrationality measure of some values of $\arctan \frac{1}{n}$
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2020
%P 32-40
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2020_12_a3/
%G ru
%F IVM_2020_12_a3
V. Kh. Salikhov; M. G. Bashmakova. On irrationality measure of some values of $\arctan \frac{1}{n}$. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2020), pp. 32-40. http://geodesic.mathdoc.fr/item/IVM_2020_12_a3/

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

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

[3] 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

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

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

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

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

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

[9] Wu Q., Wang L., “On the irrationality measure of $\log 3$”, J. number theory, 2014, no. 142, 264–273 | MR

[10] Salikhov V. Kh., Bashmakova M. G., “O pokazatele irratsionalnosti $\arctan\frac{1}{3}$”, Izv. vuzov. Matem., 2019, no. 1, 69–75 | MR | Zbl

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

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