18.785F17 Number Theory I Lecture 3 Notes: Properties of Dedekind Domains

18.785F17 Number Theory I Lecture 18 Notes: Dirichlet L-functions, Primes in Arithmetic Progressions

18.785F17 Number Theory I Lecture 19 Notes: The Analytic Class Number Formula

where a ranges over nonzero ideals of OK and p ranges over nonzero prime ideals of OK ; as we showed in the previous lecture the sum and product converge absolutely on Re(s) > 1. The following theorem is often attributed to Dirichlet, although he originally proved it only for quadratic fields (this is all he needed to prove his theorem on primes in arithmetic progressions, but we will use it in...

Lectures on Topics in Algebraic Number Theory

2 Preface During December 2000, I gave a course of ten lectures on Algebraic Number Theory at the University of Kiel in Germany. These lectures were aimed at giving a rapid introduction to some basic aspects of Algebraic Number Theory with as few prerequisites as possible. I had also hoped to cover some parts of Algebraic Geometry based on the idea, which goes back to Dedekind, that algebraic n...

18.785F17 Number Theory I Lecture 17 Notes: The Functional Equation

In the previous lecture we proved that the Riemann zeta function ζ(s) has an Euler product and an analytic continuation to the right half-plane Re(s) > 0. In this lecture we complete the picture by deriving a functional equation that relates the values of ζ(s) to those of ζ(1− s). This will then also allow us to extend ζ(s) to a meromorphic function on C that is holomorphic except for a simple ...

18.785F17 Number Theory I Lecture 16 Notes: Riemann’s Zeta Function and the Prime Number Theorem

We now divert our attention from algebraic number theory to talk about zeta functions and L-functions. As we shall see, every global field has a zeta function that is intimately related to the distribution of its primes. We begin with the zeta function of the rational field Q, which we will use to prove the prime number theorem. We will need some basic results from complex analysis, all of whic...