# Rambling Thoughts

## Dirichlet’s Theorem

We will prove Dirichlet’s Theorem on Primes in Arithmetic Progressions.

Euclid proved that there are infinitely many primes. This, however, says nothing about the distribution of those primes. Dirichlet proved a stronger statement: there are infinitely many primes in every arithmetic progression; in this sense, the primes are “uniformly distributed”.

Theorem 1 (Dirichlet’s Theorem) If ${(a, q) = 1}$, then there are infinitely many primes ${p}$ satisfying ${p \equiv a \pmod q}$.

Many ideas of the proof of Dirichlet’s Theorem were first seen in Euler’s proof of the infinitude of primes. Here, Euler actually showed that ${\sum_p 1/p}$ diverges, and that is the approach that we will take for Dirichlet’s Theorem.