**Instructor
** John
E. M^{c}Carthy

**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*