  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, Sergey Zlobin gives a change-of-variables
proof that the series and the double integral are equal.
