Honestly, I have to say that I have hardly any experience in number theory. That's maybe one additional reason why the Riemann hypothesis has such a "mystic" appearance for me. You always hear or read that it's basically "the" problem to solve in mathematics. But you always just read (as a non-mathematician) that it "has something to do with the distribution of primes".
But how do the (nontrivial) roots of $\zeta(s)$ "connect" to the distribution of primes? What's the point that makes these roots so crucial?
Thanks in advance for any answer!
EDIT: Adapted the questions, as the original ones were to broad.