Yu. V. Nesterenko

On the irrationality exponent of the number ln 2

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.

An elementary proof of the irrationality of ζ(3)

An elementary proof of the irrationality of ζ(3) is presented. The proof is based on a two times more dense sequence of Diophantine approximations to this number than the sequence in the original proof of Apery.

On Catalan’S constant

A new efficient construction of Diophantine approximations to Catalan’s constant is presented that is based on the direct analysis of the representation of a hypergeometric function with specially chosen half-integer parameters as a series and as a double Euler integral over the unit cube. This allows one to significantly simplify the proofs of Diophantine results available in this domain and substantially extend the capabilities of the method. The sequences of constructed rational approximations are not good enough to prove irrationality, but the results established allow one to compare the quality of various constructions.

On an identity of Hilbert

A simple proof of the polynomial identity used by Hilbert in the solution of the Waring problem is given. The proof is based on the continued fraction expansion of a certain formal hypergeometric series.