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.