Irrationality of Euler's constant $\gamma=0.577216...$

Question

Other constants such as $\pi$, $e$, $\phi$, $\zeta(3)$ etc, have been proven to be irrational.

There are many series, infinite products and integrals that represent Euler's constant and yet there is still the open problem of its irrationality mystery.

Why is it so hard to prove whether or not Euler's constant is irrational?

Answer 1

Other constants such as $\pi$, $e$, $\phi$, $\zeta(3)$ etc, had been proof to be of irrational constants.

Perhaps. But, in case you haven't noticed it by now, those proofs of irrationality are not one and the same. In other words, there is no catch-all method for proving that something is irrational in general. Various methods do exist for various situations $($such as the Gelfond-Schneider theorem, for instance$)$, but they do not cover all possible cases. Indeed, they don't even cover a majority of cases, but only some countable subset, whereas irrationals $($ more specifically, transcendentals $)$ are uncountable. Hope this helps.