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.

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