The Riemann Hypothesis is quite famous, but most mathematicians think about it as saying something about the Riemann zeta-function, and somehow related to primes. I will start by explaining exactly how the Riemann Hypothesis relates to the statistics of prime numbers, then go on to discuss a generalization to number fields called the Chebotarev density theorem. I'll spend the rest of my time showing how all of this generalizes to elliptic curves, and how two statements which should be equivalent aren't.