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

Salikhov, V. Kh.

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.

Reviewer: Tatiana Hessami Pilehrood (Shahrekord)

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

Keywords:

irrationality measure; linear independence measure; linear form in logarithms; Diophantine approximations

Citations:

Zbl 0632.10034; Zbl 0776.11033

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