Symmetrized polynomials in a problem of estimating of the irrationality measure of number $\ln 3$

I. V. Bondareva ; M. Yu. Luchin ; V. Kh. Salikhov

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 15-25

Résumé

An estimate of the irrationality measure of various transcendental numbers is one of the directions in the theory of Diophantine approximations foundations. Nowadays there is a range of methods which make possible to obtain similar estimates for the values of analytic functions. The most effective method is the adding of various integral constructions; one of the first early constructions is the classical intuitive representation of the Gauss hypergeometric function. Lower estimates of the irrationality measure of rational numbers logarithms were considered by many foreign authors: A. Baker and G. Wüstholz [4], A. Heimonen, T. Matala-aho, A. Väänänen [5], Q. Wu [6], G. Rhin and P. Toffin [7]. In their works they used various integral constructions, giving small linear forms from logarithms and other numbers, calculated asymptotic of integrals and coefficients of the linear forms using the saddle point method, Laplace theorem, evaluated the denominator coefficients of the linear forms using various schemes "reduction of prime numbers". Review of some methods from the theory of diophantine approximation of rational numbers logarithms at that time was introduced in 2004 by V. Zudilin [8]. Then V. Kh. Salikhov in [3] considerably improved estimate of the irrationality measure of $\ln 3$, based on the same asymptotic methods, but used a new type of integral construction, which has property of summetry. Subsequently, V. Kh. Salikhov due to usage of already complex symmetrized integral improved estimate of the irrationality measures of $\pi$ [15]. In the future, this method (as applied to diophantine approximation of logarithms of rational numbers) was developed by his pupils: E. S. Zolotuhina [10, 11], M. Yu. Luchin [12, 13], E. B. Tomashevskaya [14]. It led to improvement of the irrationality measure estimates for the following numbers: $\mu(\log(5/3))\leqslant5.512\dots$ [14], $\mu(\log(8/5))5.9897$ [12], $\mu(\log(7/5))\leqslant4.865\dots$ [14], $\mu(\log(9/7))\leqslant3.6455\dots$ [10], $\mu(\log(7/4))8.1004$ [13]. In this paper due to usage the symmetrized real integral we obtain a new estimate of the irrationality measure of $\ln 3$. The previous irrationality measure estimate of $\ln 3$ was received in 2014 by Q. Wu and L. Wang [1]. The estimate improvement had resulted from the addition of a special square symmetrized polynomial to the symmetrized polynomials used in the integral construction of K. Wu and L. Wang.

Détail

Keywords: diophantine approximations, irrationality measure, symmetrized polynomials.