Dirichlet's Theorem on Arithmetic Sequences

Theorem
Let $a, q$ be coprime integers.

Let $\mathcal P_{a, q}$ be the set of primes $p$ such that $p \equiv a \pmod q$.

Then $\mathcal P_{a, q}$ has Dirichlet density:
 * $\map \phi q^{-1}$

where $\phi$ is Euler's phi function.

In particular, $\mathcal P_{a, q}$ is infinite.

Lemma 1
Define:


 * $\eta_{a, q} : n \mapsto \begin{cases}

1 & : n \equiv a \pmod q\\ 0 & : \text{otherwise} \end{cases}$

Lemma 2
We have:

By Lemma 1, the first term grows like $\dfrac 1 {\map \phi q} \ln \dfrac 1 {s - 1}$ as $s \to 1$, while all other terms are bounded.

That is:


 * $\displaystyle \sum_{p \mathop \in \mathcal P_{a, q} } \frac 1 {p^s} \sim \frac 1 {\map \phi q} \, \map \ln {\dfrac 1 {s - 1} }$

as $s \to 1$.