## 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