This talk is an introduction to analytic number theory via two very classical results. We will start with a discussion of the Riemann zeta function, including a quick proof of the functional equation and some discussion of the location of the zeros. Then we will go through Dirichlet's proof that there are infinitely many primes congruent to a modulo $q$ when $(a, q) = 1$. To keep the talk at a survey level, some results will be stated without proof. (Translation: I'm going to avoid long, techincal calculations as much as possible.)