A Hypergeometric Approach, Via Linear Forms Involving Logarithms, to Irrationality Criteria for Euler's Constant
Abstract
Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant $\gamma$. The proof is by reduction to known irrationality criteria for $\gamma$ involving a Beukerstype double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, $\gamma$, and logarithms from Nesterenkotype series of rational functions. In the Appendix, Sergey Zlobin gives a changeofvariables proof that the series and the double integral are equal.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2002
 arXiv:
 arXiv:math/0211075
 Bibcode:
 2002math.....11075S
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Classical Analysis and ODEs;
 11J72 (Primary) 11J86;
 33C20 (Secondary)
 EPrint:
 Typos in statement of Lemma 2 corrected, reference [3] updated, published version. Appendix by Sergey Zlobin