On some methods of evaluating irrationality measure of the function $\arctan x$ values

M. G. Bashmakova ; N. V. Sycheva

Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 5-15.

Résumé

For any irrational or transcendental number estimating of the quality of its approximation by rational fractions is one of the directions in the theory of Diophantine approximations. The quantitative characteristic of such approximation is called the measure (extent) of irrationality of the number. For almost a century and a half, scientists have developed various methods for evaluating the measure of irrationality and have obtained its values for a huge number of irrational and transcendental numbers. Various approaches have been used to obtain the estimates and these approaches improved over time, leading to better estimates. The most commonly used method for obtaining such estimates is construction of linear forms with integer coefficients, which approximate a value, and studying of its asymptotic behavior. Approximating linear forms usually are constructed on the basis of continued fractions, Padé approximants, infinite series, and integrals. Methods for studying the asymptotics of such forms are currently quite standard, but the main problem is invention of a linear form with good approximating properties. The first estimates of the values of the arctangent function were obtained by M. Huttner in 1987 on the base of integral representation of the Gausss function. In 1993 A. Heimonen, T. Matala-Aho, K. Vaananen, using, like M. Huttner, Padé approximants for the Gaussian hypergeometric function, proved a general theorem for estimating of measures of irrationality of logarithms of rational numbers. Later, the same authors, using an approximating construction with Jacobi polynomials, obtained new estimates, in particular for the values of the function $\arctan x$. Further research used various integral constructions, which allowed to get both general methods for $\arctan x$ values and specialized methods for specific values. In the articles of E.B. Tomashevskaya, who in 2008 received a general estimate for the values of $\arctan\frac{1}{n}, n\in\mathbb{N}$, was used a complex integral with the property of symmetry of integrand. This property played an important role in obtaining the estimates, since it improved the asymptotic behavior of the coefficients of the linear form. Some integral constructions elaborated by other researchers also had different types of symmetry. In this article, we consider the main methods for estimating the values of the arctangent function, their features, research methods, and the best estimates at the moment.

Détail

Keywords: irrationality measure, linear form.

Bibliographie

[1] Bateman, H., Erdélyi, A., Higher transcendental functions, Mc graw-hill book company, inc., New York-Toronto-London, 1953, 456 pp. | MR | MR

[2] Huttner M., “Irrationalité de certaines integrals hypergéométriques”, Journal of Number Theory, 26 (1987), 166–178 | DOI | MR | Zbl

[3] 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

[4] Heimonen A.Matala-Aho T., Väänänen K., “An application of Jacobi type polynomials to irrationality measures”, Bulletin of the Australian mathematical society, 50:2 (1994), 225–243 | DOI | MR | Zbl

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

[6] Chudnovsky G. V., “On the method of Thue-Siegel”, Annals of math., 117:2 (1983), 325–382 | DOI | MR | Zbl

[7] Salikhov, V. Kh. Zolotukhina E. S., Bashmakova M. G., Application of symmetric integrals in the theory of Diophantine approximations, monograph, BSTU, Bryank, 2021, 124 pp. (in russian)

[8] Salikhov, V. Kh., “On the irrationality measure of $\ln 3$”, Doclady mathematics, 76:3 (2007), 955–957 | MR | Zbl

[9] Salikhov, V. Kh., “On the irrationality measure of $\pi$”, Russian Mathematical Surveys, 63:3 (2008), 570–572 | DOI | DOI | MR | Zbl | Zbl

[10] Zudilin W., Zeilbergerger D., “The Irrationality Measure of Pi is at most 7.103205334137..”, Mosc. J. of Comb. Number Theory, 9:4 (2020), 407–419 | DOI | MR | Zbl

[11] Tomashevskaya, E. B., “On the measure of irrationality of the number $\ln 5+\frac{\pi}{2}$ and some other numbers”, Chebyshevskii sbornic, 8:2 (2007), 97–108 (in russian) | MR | Zbl

[12] Tomashevskaya, E. B., On Diophantine approximations of the values of some analytic functions, dissertation for the degree of candidate of sciences - 01.01.06 “Mathematical logic, algebra and number theory”, BSTU, Bryansk, 2009, 99 pp. (in russian)

[13] Salnikova, E. S., “Diophantine approximations of $\log 2$ and other logarithms”, Mathematical Notes, 83:3 (2008), 389–398 | DOI | MR | MR | Zbl

[14] Viola C., Zudilin W., “Hypergeometric transformations of linear forms in one logarithm”, Func. Approx. Comment. Math., 39:2 (2008), 211–222 | MR | Zbl

[15] Bashmakova, M. G., “Approximation of values of the Gauss hypergeometric function by rational fractions”, Mathematical Notes, 88:6 (2010), 785–797 | DOI | DOI | MR | Zbl

[16] Bashmakova, M. G. Zolotukhina, E. S., “On irrationality measure of the numbers $\sqrt{d}\ln\frac{\sqrt{d}+1}{\sqrt{d}-1}$”, Chebyshevskii sbornic, 18:1(61) (2017), 29–43 (in russian) | DOI | MR | Zbl

[17] Bashmakova, M. G., Zolotukhina, E. S., “On estimate of irrationality measure of the numbers $\sqrt{4k+3}\ln\frac{\sqrt{4k+3}+1}{\sqrt{4k+3}-1}$ and $\frac{1}{\sqrt{k}}\arctan\frac{1}{\sqrt{k}}$”, Chebyshevskii sbornic, 19:2(66) (2018), 15–29 (in russian) | DOI | MR | Zbl

[18] Zudilin W., “One of the numbers $\zeta (5), \zeta(7), \zeta(11) $ is irrational”, Uspekhi Matematicheskikh Nauk, 56:4 (2020), 149–150 | DOI | MR

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

[20] Salikhov, V. Kh., Bashmakova, M. G., “On irrationality measure of $\arctan\frac{1}{3}$”, Russian mathematics, 2019, no. 1, 69–75 | Zbl

[21] Wu Q., Wang L., “On the irrationality measure of $\log 3$”, Journal of number theory, 142 (2014), 264–273 | DOI | MR | Zbl

[22] Salikhov, V. Kh., Bashmakova, M. G., “On irrationality measure of $\arctan\frac{1}{2}$”, Chebyshevskii sbornic, 20:4 (2019), 58–68 (in russian) | MR | Zbl

[23] Salikhov, V. Kh., Bashmakova, M. G., “On irrationality measure of some values of $\arctan\frac{1}{n}$”, Russian Mathematics, 64:12 (2020), 29–37 | MR | Zbl

[24] Wu Q., “On the linear independence measure of logarithms of rational numbers”, Math. of computation, 2002, no. 72(242), 901–911 | MR | Zbl

[25] Salikhov, V. Kh., Bashmakova, M. G., “On irrationality measure of $\arctan \frac{1}{6}, \arctan \frac{1}{10}$”, Algebra, number theory and discrete geometry: modern problems, applications and problems of history, Collection of works of XVIII international Conference, dedicated to the centenary of the birth of professors B.M.Brdikhina, V.Y. Nechaeva and S.B.Stechkina, Tolstoy Tula state pedagogical University, Tula, 2020, 264–266 (in russian)

[26] Salikhov V. Kh. Bashmakova M. G., “On rational approximations for some values of $\arctan\frac{s}{r}$ for natural $s$ and $r, s r$”, Moscow journal of combinatorics and number theory, 11:2 (2022), 181–188 | DOI | MR | Zbl