A hypergeometric approach, via linear forms involving logarithms, to criteria for irrationality of Euler's constant

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 Beukers-type 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 Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.