On astimate of irrationality measure of the logariphms of some rational numbers

M. Y. Luchin ; V. H. Salikhov ; E. S. Zolotukhina

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 226-235

Résumé

At present the use of symmetrized integrals is one of the approaches to obtain estimates of irrationality measures. They were considered in the past (see [1]), but the most dynamic development of this direction acquired in the articles of V. H. Salikhov and his students (see [2]–[5]). The beginning was V. H. Salikhov's article [6], in which the estimate of irrationality measure of the number $\ln{3}$ was improved: $\mu(\ln{3})\leq5.125$. In 2014 Q. Wu and L. Wang improved on this result and received the estimate $\mu(\ln{3})\leq5.1163051$. In their work symmetrized polynomials of the first degree were used. With help of the integral construction based on symmetrized polynomials of first and second degree I. V. Bondareva, M. Y. Luchin and V. H. Salikhov in [8] improved the result of Q. Wu and L. Wang: $\mu(\ln{3})\leq5.116201$. For the first time quadratic symmetrized polynomials were used in [9]. Using similar polynomials in the complex integral (modified Tomashevskay integral) V. H. Salikhov and E. S. Zolotukhina improved the estimate of the irrationality measure of the number $\ln{\frac{5}{3}}$: $\mu(\ln{\frac{5}{3}})\leq5.119417\ldots$. Previous estimates were found by E. B. Tomashevskay [11], E. S. Zolotukhina [12], К. Väänänen, А. Heimonen, Т. Matala-aho [13]. The aim of this article is to obtain a new estimate of join approximations by the numbers $1,\ \ln{2},\ \ln{3},\ \ln{5}$ and $1,\ \ln{2},\ \ln{3},\ \ln{5},\ \ln{7}$ using polynomial of first and second degree in the integral construction.

Détail

Keywords: irrationality measure, join approximations, symmetrized integrals.