## On the irrationality measure of $$\ln 3$$.(English. Russian original)Zbl 1169.11032

Dokl. Math. 76, No. 3, 955-957 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 417, No. 6, 753-755 (2007).
The best bound for the irrationality measure of $$\ln 3$$ known so far was obtained by G. Rhin [Théorie des Nombres, Sémin. Paris 1985/1986, Prog. Math. 71, 155–164 (1987; Zbl 0632.10034)], who proved that $$\mu(\ln 3)\leq 8.616.$$ In the paper under review, the author significantly improves this estimate and establishes the new record bound $$\mu(\ln 3)\leq 5.125.$$ The main result of the paper is as follows.
{Theorem.} Suppose that $$q, p_1, p_2\in \mathbb Z$$ and $$Q=\max(|q|, |p_1|, |p_2|),$$ $$Q\geq Q_0,$$ where $$Q_0$$ is a sufficiently large number. Then $|q+p_1\ln 2+p_2\ln 3|>\frac{1}{Q^{4.125}}.$ As a consequence, one gets that $$|\ln 3 -\frac{p}{q}|>\frac{1}{q^{5.125}}$$ for any $$p, q\in {\mathbb Z}$$ with $$q\geq q_0.$$ The proof essentially uses M. Hata’s construction [Acta Arith. 63, No. 4, 335–349 (1993; Zbl 0776.11033)], but with a different integral. The author considers the integrals $$I(\alpha)=\int_{35}^{\alpha}R(x)\,dx,$$ where $R(x)=\frac{(x-28)^n(x-30)^n(x-35)^{2n}(x-40)^n(x-42)^n}{x^{2n+1}(70-x)^{2n+1}},$ where $$\alpha\in\{40, 42\}$$ and $$n$$ is an even positive integer. The following property of the integrand $$R(x)$$ is crucial: $$R(70-x)=R(x).$$ This allows the author to show that $$35I(40)\cdot 2^{n+1}q_{2n}=A\ln\frac{4}{3}+B_1$$ and $$35I(42)\cdot 2^{n+1}q_{2n}=A\ln\frac{3}{2}+B_2,$$ where $$A, B_1, B_2\in {\mathbb Z},$$ $$q_{2n}= \text{lcm} (1,2,\dots, 2n).$$ The exact asymptotics of $$A, I(40), I(42)$$ as $$n\to\infty$$ is calculated by the saddle point method.

### MSC:

 11J82 Measures of irrationality and of transcendence 11J86 Linear forms in logarithms; Baker’s method 11J20 Inhomogeneous linear forms 11K60 Diophantine approximation in probabilistic number theory

### Citations:

Zbl 0632.10034; Zbl 0776.11033
Full Text:

### References:

 [1] G. Rhin, in Seminaire De Theorie Des Nombres (Paris, 1985–1986), Ed. by C. Golgstein (Birkhäuser, Boston, 1987), pp. 155–164. [2] M. Hata, Acta Arith. 63(4), 335–349 (1993). · Zbl 0776.11033
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