A family of criteria for irrationality of Euler's constant

Following earlier results of Sondow, we propose another criterion of irrationality for Euler’s constant $\gamma$. It involves similar linear combinations of logarithm numbers $L\mathrm{\_}n,m$. To prove that $\gamma$ is irrational, it suffices to prove that, for some fixed $m$, the distance of $d\mathrm{\_}nL\mathrm{\_}n,m$ ($d\mathrm{\_}n$ is the least common multiple of the $n$ first integers) to the set of integers $\mathbf{Z}$ does not converge to 0. A similar result is obtained by replacing logarithms numbers by rational numbers: it gives a sufficient condition involving only rational numbers. Unfortunately, the chaotic behavior of $d\mathrm{\_}n$ is an obstacle to verify this sufficient condition. All the proofs use in a large manner the theory of Pad'e approximation.