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.
Basis for Grading
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