Criteria for irrationality of Euler's constant

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 with d(n) = LCM(1,..., n) the following are equivalent: 1. The fractional part of log S-n is given by {log Sn} = d(2n)I(n) for some n. 2. The formula holds for all sufficiently large n. 3. Euler's constant is a rational number. A corollary is that if {log S-n} greater than or equal to 2(-n) infinitely often, then gamma is irrational. Indeed, if the inequality holds for a given n ( we present numerical evidence for 1 < n less than or equal to 2500) and gamma is rational, then its denominator does not divide d(2n)((2n)(n)). We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact log S-n. A by-product is a rapidly converging asymptotic formula for gamma, used by P. Sebah to compute gamma correct to 18063 decimals.