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

Abstract

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, Pad´e 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, Pad´e 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 1/𝑛, 𝑛 ∈ 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.