On irrationality measure of $\ln{\frac{5}{3}}$

V. Kh. Salikhov ; E. S. Zolotukhina ; E. B. Tomashevskaya

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 330-338

Résumé

In this paper the estimation of irrationality measure of $\ln{\frac{5}{3}}$ is refined. To date, a lot of estimates of irrationality measures for the values of analytic functions have been established, in particular, logarithms of rational numbers. Diophantine approximations of logarithms of rational numbers were considered in the papers of K. Vaananen, A. Heimonen, T. Matala-aho [1], G. Rhin [2], Е. А. Rukhadze [3], М. Hata [4]–[6] and other. Тhese authors used integral constructions that give small linear forms from the numbers and have good estimates of the denominators of the coefficients. Аsymptotics of integrals and the coefficients of the linear forms computed by using theorem of Laplace and the method of the pass. An overview of some constructions from the theory of Diophantine approximations of logarithms of rational numbers was presented in the article by V. V. Zudilin. Note that in 2009 R. Marcovecchio with the help of the double complex integral has received the best estimate of the irrationality measure of $\ln{2}$. Recently, the symmetries of functions involved in integral constructions are often used. The use of symmetrized integrals allowed E. Zolotukhina in [9] and E. Tomashevskay in [10] to obtain new estimates of irrationality measures of some logarithms of rational numbers. For the first time such an integral was considered by V. H. Salikhov in obtaining an estimate of the irrationality measure of $\ln3$ in [11] and $\pi$ in [12]. In 2014, Q. Wu and L. Wang in [13] received an estimate of the irrationality measure of $\ln{3}$, which improved V. H. Salikhov's result. For the first time in their work, general symmetrized polynomials of the first degree of the form $At-B$, $t=(x-d)^2$, were applied. In 2017, V. H. Salikhov, I. Bondareva and M. Luchin in [14] improved Q. Wu's result on the irrationality measure of $\ln{7}$ (see [15]). Here was first considered the quadratic symmetrized polinomials. In this paper, quadratic symmetrized polynomials are also used, but a complex integral will be considered.

Détail

Keywords: Irrationality measure, symmetrized integrals, symmetrized polinomials.