Why L-functions are Important?

In this note, we will prove the special case of Dirichlet's theorem on arithmetic progression, by stating that there exists infinitely many primes of the forms $4k+1$ and $4k+3$. It is not difficult to proving this fact by elementary number theory. But this method cannot lead us to go further to predict that there exists finite or infinite primes of the form $37k + 19$, for example.

Dirichlet proved his famous theorem by using analytic theory of L-functions associated with Dirichlet's characters. The proof is long, and difficult, with quite deep consideration on complex theory of arithmetical functions. This short note give an elementary, and motivational introduction to this powerful method.