Approximation of $\ln{\frac{\sqrt{5}-1}{2}}$ by numbers of the field $\mathbb Q\left(\sqrt{5}\right)$

V. Kh. Salikhov ; E. S. Zolotukhina

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 339-356

Résumé

The article continues the study of the integral construction, which was first considered by V. H. Salikhov and V. A. Androsenko in 2015 [1]. This construction is a modification of the integral that was introduced by R. Marcovecchio in 2009 to find the irrationality measure of $\ln{2}$. With the help it V. A. Androsenko improved the estimate of irrationality measure of $\frac{\pi}{\sqrt{3}}$ in [1]. The previous results belonged to the L.V. Danilov [3], K. Aladi and M Robinson [4], G. V. Chudnovsky [5], А. К. Dubickas [6], M. Hata [7], [8], G. Rhin [9]. Another direction of the study of this integral construction is to obtain estimates of the approximation of some constants by numbers from quadratic fields. In 2016 M.Y.Luchin and V. H. Salikhov improved the estimate of the approximation of $\ln{2}$ by the numbers of the field $\mathbb{Q}(\sqrt{2})$. Previous estimates were found by F. Amoroso F. and C. Viola [11] and E. S. Zolotukhina [12]. The aim of this article is to obtain a new estimate of the approximation of logarithm of “Golden section” by the number of the field $\mathbb{Q}(\sqrt{2})$. Previous estimates were found by V. H. Salikhov and E. S. Zolotukhina [13].

Détail

Keywords: irrationality measure, quadratic irrationalities, symmetrized integrals.