Cornell Math - MATH 735, Spring 2000

MATH 735 — Spring 2000
Introduction to Analytic Number Theory

This course will be an introduction to the study of prime numbers and arithmetical functions. It is intended for students with little or no prior exposure to these topics and will include proofs of the fundamental results, such as the Prime Number Theorem (PNT) and Dirichlet's theorem on primes in an arithmetic progression, as well as much else.

Of the numerous methods available for the study of the natural numbers, we shall discuss at least three: elementary methods, sieve methods and the theory of the Riemann zeta function and related functions. Others, such as van der Corput's method and the Hardy-Littlewood circle method, may also be included. The elementary methods grew out of Chebychev's unsuccessful attempts to prove PNT. They are appealing because they relate in a direct way to the questions they are intended to answer. They are also surprisingly powerful; powerful enough to prove PNT, for example. Sieve methods have their origin in the Sieve of Eratosthenes. In their modern incarnation they are methods for estimating the size of a set of integers given some information about how that set is distributed among congruence classes. The study of the Riemann zeta function led to the first proof of PNT and remains essential in understanding the finer properties of the distribution of primes, as well as in many other arithmetical questions.

At first, the required background knowledge will be minimal. For our discussion of elementary and sieve methods only the basic facts of arithmetic and some elementary analysis (mostly calculus) will be needed. Later on, a facility with the basics of complex analysis and possibly also of Fourier analysis, plus a willingness to pick up some further results from those subjects as we go along, would suffice.

There will be no textbook for the course, but Davenport's "Multiplicative Number Theory" would be a suitable reference for a substantial part of it. Perhaps even more than most mathematics, analytic number theory makes an unsatisfying spectator sport. The analytic detail of the arguments can easily obscure the main lines of thought and the subject also has its share of unfamiliar notation. To alleviate these problems, homework exercises will be suggested from time to time. These will be optional, but anyone who wants to follow the discussion will probably have to solve at least some of them.