On irrationality measure $\mathop{\mathrm{arctg}}\frac{1}{2}$

M. G. Bashmakova ; V. Kh. Salikhov

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 58-68.

Résumé

An evaluation of irrationality measure for various transcendental numbers is one of the field in diophantine approximation theory. Starting with the works of Е. Borel at the end of 19th century, were developed both general methods of evaluation for classes of some functions values and specialized approaches for estimating peculiar numbers. Diverse methods particularly were practiced for the investigating of arithmetic properties of the function $\mathop{\mathrm{arctg}} x$ values. For getting evaluation on irrationality measure of $\mathop{\mathrm{arctg}}x $ values many authors regarded them as particular case of Gauss hypergeometric function. One of the first such kind of papers was the article of M. Huttner 1987 [1], who proved a generalized theorem about estimation on irrationality measure of the Gauss hypergeometric function values $F_2^1\left(1,\frac{1}{k},1+\frac{1}{k}|\varepsilon x^k\right), k\in\mathbb N, k\geq 2, \varepsilon=\pm 1.$ A big role in progress of theme have been played by works of A. Heimonen, T. Matala-aho, K. Väänänen [2], [3], in which was also constructed a method for evaluation on irrationality measure of the Gauss hypergeometric function values of the form $F_2^1\left(1,\frac{1}{2},1+\frac{1}{2}|z\right), k\in \mathbb N, k\geq 2$, including $F_2^1\left(1,\frac{1}{2},\frac{3}{2}|-z^2\right)=\frac {1}{z}\mathop{\mathrm{arctg}} z.$ The approach considered by them had used approximation of the Gauss hypergeometric function by Jacobi type polynomials and gave a lot of concrete results. Last decades for evaluation of various numbers were broadly spreading methods, which used symmetric on some changes of variable integrals [4], [5], [6]. Originally, integral qualitatively using the property of symmetry was applied by V.Kh.Salikhov [4], who used it to got the new estimate for $\ln 3.$ A little later V. Kh. Salikhov [7] had applied similar symmetrized complex integral for obtaining new evaluation of $\pi.$ In that work he put to use classical equality $\frac{\pi}{4}=\mathop{\mathrm{arctg}} \frac{1}{2}+\mathop{\mathrm{arctg}} \frac{1}{3}.$ The same method, i.e. complex symmetrized integral was used by E. B. Tomashevskaya [8], who had estimated values of $\mathop{\mathrm{arctg}} \frac{1}{n}, n\in\mathbb N, n>2$ and some of previous results for such numbers were improved by her. Later on E. B. Tomashevskaya [9] had elaborated analogical integral for estimation of $\mathop{\mathrm{arctg}}\frac{1}{2}$, which one had allowed to prove the best result until now $\mu(\mathop{\mathrm{arctg}} \frac{1}{2})\leq 11.7116\dots$. In 2014 K. Wu and L. Vang [10] improved the result of V. Kh. Salikhov for $\ln 3$, applying a new type integral construction, which also had used a property of symmetry. In present paper we took the idea of K. Wu and L. Vang and applied it to the integral of E. B. Tomashevskaya. It allowed us to improve arithmetic properties of integral and obtain better result for extent of irrationality $\mathop{\mathrm{arctg}}\frac{1}{2}$.

Détail

Keywords: irrationality measure, hypergeometric function, symmetrized integral.

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