Soft question: Where is the connection between the zeros of Riemann's $\zeta$-funciton and the density of prime numbers?
Is there a short answer to this question, to get the overview? I once had a lecture about this topic, but it took several weeks to proof the prime number theorem using $\zeta$, so I lost track of the general idea.
The Riemann hypothesis, about the non-trivial zeros of $\zeta(s)$, is equivalent to the following statement: For any real number $x\ge 1$ the number of prime numbers less than $x$ is approximately $Li(x)$ and this approximation is essentially square root accurate. More precisely, $$ \pi(x)=Li(x)+O(\sqrt{x}\log(x)). $$ Here $\pi(x)=\sum_{p\le x}1$ is the number of primes up to $x$. This says that the distribution of primes has the best possible error term if and only if the Riemann hypothesis holds true.