Let $$\pi(x)$$ be the number of prime numbers $$< x \in \mathbb{Z}$$. Studying the asymptotics of $$\pi(x)$$ already asks how it grows as $$x$$ goes to infinity. Euclid gave an answer to this but didn't finish the story. Analysis, the study of functions, convergence, and calculus, plays a big role in number theory, from finishing Euclid's story to proving Dirichlet's theorem on arithmetic progressions, and even another proof of quadratic reciprocity. We'll introduce how analysis becomes an essential tool, and talk about problems that are still out there for people to solve. Any remaining time will be allotted for questions about any math.