Of course, we've not found a closed-form solution for the Riemann Zeta function; one that only involves 'elementary functions' (and not recursion, like Riemann's functional equation). However, I'm wondering if we know for certain that this is impossible. If so, how?
In general, is there any way of knowing if it would be possible or not to find such a solution? Or at least, some clues before we waste time checking?