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Chebyshevskii Sbornik, 2019, Volume 20, Issue 4, Pages 339–356
DOI: https://doi.org/10.22405/2226-8383-2018-20-4-339-356
(Mi cheb852)
 

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

Bryansk State Technical University
References:
Abstract: 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].
Keywords: irrationality measure, quadratic irrationalities, symmetrized integrals.
Funding agency Grant number
Russian Foundation for Basic Research 18-01-00296_а
The work was carried out with the financial support of the RFBR, grant №18-01-00296 А.
Received: 25.06.2019
Accepted: 20.12.2019
Document Type: Article
UDC: 511.36
Language: Russian
Citation: V. Kh. Salikhov, E. S. Zolotukhina, “Approximation of $\ln{\frac{\sqrt{5}-1}{2}}$ by numbers of the field $\mathbb Q\left(\sqrt{5}\right)$”, Chebyshevskii Sb., 20:4 (2019), 339–356
Citation in format AMSBIB
\Bibitem{SalZol19}
\by V.~Kh.~Salikhov, E.~S.~Zolotukhina
\paper Approximation of $\ln{\frac{\sqrt{5}-1}{2}}$ by numbers of the field $\mathbb Q\left(\sqrt{5}\right)$
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 4
\pages 339--356
\mathnet{http://mi.mathnet.ru/cheb852}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-4-339-356}
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