Representation theory: Dirichlet's theorem
Channel: Richard E Borcherds
Date: 2020-09-08

In this talk we see how to use characters of finite abelian groups to prove Dirichlet's theorem that there are infinitely many primes in certain arithmetic progressions. We first recall Euler's proof that there are infinitely many primes, which is the simplest case of Dirichlet's proof. 
Then we define Dirichlet characters and Dircihlet L-functions and give some examples. 

We prove Dirchlets theorem for the special case of primes that are 3 mod 8 
by using the orthogonality relations of Dircihlet characters, together with the fact that Dirichlet L-series L(s,X) do not vanish at s=1.

Finally we sketch how to use the orthogonality relations to show the non-vanishing of Dirichlet characters at s=1.