Symmetrized polynomials in a problem of estimating of the irrationality measure of number ln3

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: А. Baker and G. Wüstholz [4], А. Heimonen, Т. Matala-aho, К. Vaänanen [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 ln3, 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 π [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], Е. B. Tomashevskaya [14]. It led to improvement of the irrationality measure estimates for the following numbers:

µ(log(5/3)) ≤ 5.512... [14], µ(log(8/5)) < 5.9897 [12], µ(log(7/5)) ≤ 4.865... [14], µ(log(9/7)) ≤6 3.6455... [10], µ(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 ln3. The previous irrationality measure estimate of ln3 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.

1. Wu, Q. & Wang, L. 2014, “On the irrationality measure of log3“, Journal of Number Theory., vol. 142, pp. 264-273.

2. Rhin, G. 1987, “Approximants de Pade et mesures effectives d’ irrationalite, Seminaire de Theorie des Nombres, Paris 1985-1986“, Progress i Math., Boston: Birkhauser., no. 71, pp. 155164.

3. Salikhov, V. H. 2007, “On the measure of irrationality ln3“, Doklady Mathematics, vol. 47, no. 6, pp. 753-755.

4. Baker, A. Wüstolz G. 1993, “Logarithmic forms and group varieties“, J. Reine Angew. Math., vol. 442, pp. 19-62.

5. Heimonen, A. & Matala-aho, T. Väänänen K. 1993, “On irrationality measures of the values of Gauss hypergeometric function“, Manuscripta Math., vol. 81, no. 1, pp. 183-202.

6. Wu, Q. 2002, “On the linear independence measure of logarithms of rational numbers“ ,Math. Comput., vol. 72, no. 242, pp. 901-911.

7. Rhin, G. & Toffin, P. 1986, “Approximants de Padé simultanés de logaritmes“, J. Number Theory., vol. 24, pp. 284-297.

8. Zudilin, V. V. 2004, “An essay on irrationality measures of π and other logarithms“, CHebyshevskij sbornik, vol. 5, no. 2, pp. 49-65.

9. Salikhov, V. H. 2010, “On the measure of irrationality of a number π“, Mathematical Notes, vol. 88, no. 4, pp. 583-593.

10. Zolotukhina, Е. S. 2009, “Diophantine approximations of some logarithms: dis. kand. fiz.-mat. nauk“, Bryansk, 100 p.

11. Sal’nikova, Е. S. 2010, “Approximations of some logarithms by numbers from fields Q и Q(√d)“, Fundam. i prikl. matematika, vol. 16, no. 6, pp. 139-155.

12. Luchin, M. Y. 2012, “On Diophantine approximations of some logarithms “ Vestnik Bryanskogo gosudarstvennogo universiteta , no. 4 (2), pp. 22-28.

13. Luchin, M. Y. 2013, “Measure of irrationality of a number 7/4 “ CHebyshevskij sbornik , vol. 14, no. 2, pp. 123-1

14. Tomashevskaya, Е. B. 2010, “On Diophantine approximations of function values logx“ Fundam. i prikl. matematika , vol. 16, no. 6, pp. 157-166.

15. Hata, M. 1993, “Rational approximations to π and some other numbers“, Acta Arith., vol. LXIII, no. 4, pp. 335-349.