The estimate of the irrationality measure of number $\log\dfrac{37}{30}$

M. Yu. Luchin

Čebyševskij sbornik, Tome 15 (2014) no. 1, pp. 131-140.

Résumé

Lower estimates of the irrationality measure of logarithms of rational numbers considered by many foreign authors: M. Waldschmidt [1], A. Baker and G. Wüstholz [2], A. Heimonen, T. Matala-aho, K. Väänänen [3], Q. Wu [4], G. Rhin [5] and P. Toffin [6]. 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 logarithms of rational numbers at that time was introduced in 2004 by V. Zudilin [7]. Then V. Kh. Salikhov in [8] considerably improved estimate of the irrationality measure of $\log3$, based on the same asymptotic methods, but used a new type of integral construction, which has property of symmetry. Subsequently, V. Kh. Salikhov due to usage of already complex symmetrized integral improved estimate of the irrationality measures of $\pi$ [9]. 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 estimates of the irrationality measure following numbers: \begin{gather*} \mu{(\log{(5/3)})}\leq\:5.512...~[14],\quad \mu{(\log{(8/5)})}\:\:5.9897~[12],\quad \mu{(\log{(7/5)})}\leq\:4.865...~[14],\\ \mu{(\log{(9/7)})}\leq\:3.6455...~[10],\quad \mu{(\log{(7/4)})}\:\:8.1004~[13]. \end{gather*} In this paper due to usage the symmetrized real integral we obtain a new estimate of the irrationality measure of $$\tau=\log{(37/30)},\quad \mu{(\tau)}\:\:65.3358.$$ First time estimate of the irrationality measure of $\log{(37/30)}$ was received in 1993 by A. Heimonen, T. Matala-aho, K. Väänänen [1]. In their work they received a common criterion that allows to evaluate irrationality measure of numbers of the form $\log({1-(r/s))}$, where $r/s\in[-1,~1)$ $(r, s\in\mathbb{N})$. As an example, they led a table with the resulting estimates at some values $r/s$. One of these values was the number $r/s=-7/30$, which gave following estimate: $\mu{(\log{(37/30)})}\leq\:619.5803\dots$ We also note, that for obtain a new estimate the optimal parameters of integral construction were calculated using the developed by the author of a computer program, which uses the Mathcad calculations.

Détail

Keywords: diophantine approximations, irrationality measure, saddle point method.

Bibliographie

[1] Waldschmidt M., “Minorations de combinaisons linéaires de logarithmes de nombres algébriques”, Can. J. Math., 45:1 (1993), 176–224 | DOI | MR | Zbl

[2] Baker A., Wüstholz G., “Logarithmic forms and group varieties”, J. Reine Angew. Math., 442 (1993), 19–62 | MR | Zbl

[3] Heimonen A., Matala-aho T., Väänänen K., “On irrationality measures of the values of Gauss hypergeometric function”, Manuscripta Math., 81:1 (1993), 183–202 | DOI | MR | Zbl

[4] Wu Q., “On the linear independence measure of logarithms of rational numbers”, Math. Comput., 72:242 (2002), 901–911 | DOI | MR

[5] Rhin G., “Approximants de Padé et mesures effectives d'irrationalité”, Séminaire de Théorie des Nombres (Paris 1985–1986), Progress in Math., 71, 1987, 155–164 | DOI | MR | Zbl

[6] Rhin G., Toffin P., “Approximants de Padé simultanés de logarithmes”, J. Number Theory, 24 (1986), 284–297 | DOI | MR | Zbl

[7] Zudilin V. V., “Esse o merakh irratsionalnosti $\pi$ i drugikh logarifmov”, Chebyshevskii sbornik, 5:2 (2004), 49–65 | MR | Zbl

[8] Salikhov V. Kh., “O mere irratsionalnosti $\ln3$”, DAN RF, 417:6 (2007), 753–755 | MR | Zbl

[9] Salikhov V. Kh., “O mere irratsionalnosti chisla $\pi$”, Matematicheskie zametki, 88:4 (2010), 583–593 | DOI | MR | Zbl

[10] Zolotukhina E. S., Diofantovy priblizheniya nekotorykh logarifmov, dis. ... kand. fiz.-mat. nauk, Bryansk, 2009, 100 pp.

[11] Salnikova E. S., “Priblizheniya nekotorykh logarifmov chislami iz polei $\mathbb{Q}$ i $\mathbb{Q}(\sqrt d)$”, Fundam. i prikl. matematika, 16:6 (2010), 139–155 | MR

[12] Luchin M. Yu., “O diofantovykh priblizheniyakh nekotorykh logarifmov”, Vestnik Bryanskogo gosudarstvennogo universiteta, 2012, no. 4(2), 22–28

[13] Luchin M. Yu., “Otsenka mery irratsionalnosti chisla $\ln\frac{7}{4}$”, Chebyshevskii sbornik, 14:2 (2013), 123–131

[14] Tomashevskaya E. B., “O diofantovykh priblizheniyakh znachenii funktsii $\log x$”, Fundam. i prikl. matematika, 16:6 (2010), 157–166 | MR

[15] Hata M., “Rational approximations to $\pi$ and some other numbers”, Acta Arith., 63:4 (1993), 335–349 | MR | Zbl