Math 521, Spring 2015
Dirichlet Series

Instructor          John E. McCarthy
Class                 MWF 2.00-3.00, Cupples I Rm 215
Office                105 Cupples I
Office Hours     Come to my office any afternoon.
Phone                 935-6753

Prerequisites

Complex Analysis 5021-5022 and Real Analysis 5051-5052.

Content

This is a course about Dirichlet Series. I will assume that you know basic functional analysis (as covered, for example, in Folland's "Real Analysis" book) and the complex analysis of 5021-5022. Familiarity with Hardy spaces would be an asset, but is not essential.

I shall start out defining Dirichlet series - functions of the form $f(s) = \sum a_n n^{-s}$. We shall look at some convergence issues, and then we shall move on to looking at Banach spaces of Dirichlet series. A central theme in the course will be how Dirichlet series, even though they are a function of only one variable, behave in many ways like a power series in infinitely many variables.

Dirichlet series were introduced to study prime numbers. There are obviously infinitely many primes whose residue mod 4 is either 1 or 3, but how do you show that both these classes are infinite?
We shall also look at some number theory that uses the theory of Dirichlet series.

Attendance and class participation.

Bibliography

The best book on the subject is  Diophantine approximation and Dirichlet series, by H. and M. Queff\'elec.
Other useful viewpoints are:

H. Helson                      Dirichlet Series
E.C. Titchmarsh            The theory of the Riemann zeta-function

For background on the real and complex analysis underpinnings, there are many choices. Here are two.

P. Duren                      Theory of H^p-spaces
W. Rudin                      Real and Complex Analysis