I saw one of the expansions of Euler Mascheroni constant in terms of Meissel Mertens constant as a consequence of Mertens theorem.
$$ B = \gamma + \sum_p \left\{ \log\left( 1 - \frac 1p\right) + \frac 1p\right\}$$
This is that expansion. Now I don't understand why is it difficult to prove the irrationality of Euler Mascheroni constant. Since we have infinitely many prime numbers, the sum over all those primes in the above equation, if converges must be a irrational, then why is it not considered as a proof of irrationality?