Relation between the prime density and Riemann's zeros?

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.

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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.
Well, more or less the same ideas as for the PNT, which took several weeks, as you said. Elkies has a relatively short proof, see page $4$ in math.harvard.edu/~elkies/M259.02/pnt.pdf. –  Dietrich Burde Nov 26 '13 at 19:39