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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
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[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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