  By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we
derive criteria for irrationality of Euler's constant, gamma. For n > 0, we
define a double integral I(n) and a positive integer S(n), and prove that if
d(n) = LCM(1,...,n), then the fractional part of logS(n) is given by {logS(n)}
= d(2n)I(n), for all n sufficiently large, if and only if gamma is a rational
number. A corollary is that if {logS(n)} > 1/2^n infinitely often, then gamma
is irrational. Indeed, if the inequality holds for a given n (we present
numerical evidence for 0 < n < 2500 and n = 10000) and gamma is rational, then
its denominator does not divide the product d(2n)Binomial(2n,n). We prove a new
combinatorial identity in order to show that a certain linear form in
logarithms is in fact logS(n). A by-product is a rapidly converging asymptotic
formula for gamma, used by P. Sebah to compute it correct to 18063 decimals.
