"We will survey some topics related to the ""amount"" of prime numbers and approaches to this kind of question using analytic and topological ideas. For example, one can prove that there are infinitely many primes using analytic tools (the zeta function) or topology (taking the profinite completion of the integers). Similarly, one can look for a topological version of Dirichlet's theorem that there are infinitely many primes in arithmetic progressions. The talk should be largely elementary and accessible to undergraduates. NoteThis is not relly a discrete mathematics talk, but will be accessible and interesting to all. "