# Analytic Number Theory

Analytic Number Theory (Spring 2019) is about the use of calculus in studying the positive integers. At first, this sounds paradoxical: how can a continuous tool like calculus be useful for studying discrete objects like the integers? Nonetheless, it turns out that calculus is an extremely powerful tool for studying the integers.

A typical question in analytic number theory is to determine the number of prime numbers up to some bound $x$. Maybe not the exact number, but a pretty good approximation. The most famous result along these lines is the prime number theorem, which says that the number of primes up to $x$ is $\frac{x}{\log(x)}$.

Another major result in analytic number theory is Dirichlet’s Theorem on primes in arithmetic progression, which says that if $\gcd(a,m)=1$, then there are infinitely many primes congruent to $a\pmod{m}$. We will study this, and along the way, we will learn a lot about Dirichlet characters and $L$-functions.

Another sort of question for which analytic questions are amenable is the problem of determining average sizes of arithmetic functions. For instance, how many divisors does a typical number of size around $x$ have? There is a general method involving Dirichlet series for approaching this problem, as well as more hands-on methods, involving the classic but extremely powerful technique of switching the order of summation.

Knowledge of calculus is assumed in this class.

This class will begin on Tuesday, April 2nd, and will meet for 10 consecutive Tuesdays and Wednesdays from 6:30–8:30 PM in Palo Alto. The class will end on June 5th.