Rational approximations to \(\pi\) and some other numbers. (English) Zbl 0776.11033

Hata, Masayoshi

Acta Arith. 63, No. 4, 335-349 (1993).

The author proves the irrationality measure 4.6016 for \(n/\sqrt 3\) and 8.0161 for \(\pi\) and \(\pi/\log 2\). The proof is based on the expression of the integral \[ \int_ \Gamma ((z-a_ 1)^ 2(z-a_ 2)^ 2(z-a_ 3)^ 2/z^ 3)^ n dz/z \] as a sum of simple integrals for suitable numbers \(a_ 1\), \(a_ 2\) and \(a_ 3\). Similarly the author obtains the linear independence measure for 1, \(n\sqrt 3\), \(\log(3,4)\), and some other results. Let us note, that the above measures contain roots of cubic equations with integral coefficients.

Reviewer: J.Hančl (Ostrava)

MSC:

11J82	Measures of irrationality and of transcendence
11J72	Irrationality; linear independence over a field

Keywords:

irrationality measure; linear independence measure