On the irrationality exponent of the number ln 2

@article{Nesterenko2010OnTI,
  title={On the irrationality exponent of the number ln 2},
  author={Yu. V. Nesterenko},
  journal={Mathematical Notes},
  year={2010},
  volume={88},
  pages={530-543}
}
  • Y. Nesterenko
  • Published 9 November 2010
  • Mathematics
  • Mathematical Notes
We propose another method of deriving the Marcovecchio estimate for the irrationality measure of the number ln 2 following, for the most part, the method of proof of the irrationality of the number ζ(3) proposed by the author in 1996. The proof uses single complex integrals, the so-called Meyer G-functions, and is much simpler than that of Marcovecchio. 

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References

SHOWING 1-10 OF 18 REFERENCES

A few remarks on ζ(3)

A new proof of the irrationality of the number ζ(3) is proposed. A new decomposition of this number into a continued fraction is found. Recurrence relations are proved for some sequences of

Approximations to the logarithms of certain rational numbers

In a recent paper [1] methods were introduced for investigating the accuracy with which certain algebraic numbers may be approximated by rational numbers. It is the main purpose of the present paper

The group structure for ζ(3)

1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. ..

Number Theoretic Applications of Polynomials with Rational Coefficients Defined by Extremality Conditions

It is well known that classes of polynomials in one variable defined by various extremality conditions play an extremely important role in complex analysis. Among these classes we find orthogonal

A Course of Modern Analysis

The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.

Approximants de Padé et mesures effectives d’irrationalité

Les approximants de Pade; des fonctions hypergeometriques ont ete utilises pour l’etude en des points rationnels z=p/q des approximations diophantiennes des valeurs de ces fonctions. Cette methode

Legendre type polynomials and irrationality measures.

The Rhin-Viola method for log 2