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The Rhin-Viola method for \(\log 2\). (English) Zbl 1197.11083

The author gives upper bound for irrationality measure and non-quadraticity measure of logarithms of rational numbers. In particular, the author gives bounds \(\mu(\log 2)<3.5746\) for irrationality measure and \(\mu_2(\log 2)<15.6515\) for non-quadraticity measure.
The author introduces a family of double complex integrals depending on six parameters. Combining methods of Sorokin and Rhin and Viola he finds a permutation group acting on the set of parameters such that the integrals are invariant under the action of that group. The integrals can be expressed as forms in \(1\) and logarithms where the coefficients are polynomials with rational coefficients. The result is obtained from the asymptotic behaviour of those polynomials and from results of Hata.

MSC:

11J82 Measures of irrationality and of transcendence
11J04 Homogeneous approximation to one number

Citations:

Zbl 1201.11073
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