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:
 * $\phi \left({q}\right)^{-1}$

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

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

Lemma 1
Let $\chi$ be a Dirichlet character modulo $q$.

Let:
 * $\displaystyle f \left({s}\right) = \sum_p \chi \left({p}\right) p^{-s}$

If $\chi$ is non-trivial then $f \left({s}\right)$ is bounded as $s \to 1$.

If $\chi$ is the trivial character then:


 * $f \left({s}\right) \sim \ln \left({\frac 1 {s-1} }\right)$

as $s \to 1$.

Proof of Lemma 1
By Logarithm of Dirichlet L-Functions:


 * $(1):\quad \displaystyle \sum_p \chi \left({p}\right) p^{-s} = \ln L \left({s, \chi}\right) - \sum_p \sum_{n \ge 2} \frac{\chi \left({p}\right)^n} {n p^{n s}}$

If $\chi$ is non-trivial, then by L-Function does not Vanish at One, $\ln L \left({s, \chi}\right)$ is bounded as $s \to 1$.

If $\chi$ is trivial, then by Analytic Continuation of Dirichlet L-Function, $L \left({s, \chi}\right)$ has a simple pole at $s = 1$.

Therefore, in this case:
 * $L \left({s, \chi}\right) \sim \dfrac \lambda {s-1}$

where $\lambda$ is the residue of $L \left({s, \chi}\right)$ at $1$, and:


 * $\ln L \left({s, \chi}\right) \sim \ln \left({\dfrac \lambda {s-1}}\right) \sim \ln \left({\dfrac 1 {s-1}}\right)$

Thus if we can show that the second term of $(1)$ is bounded, the result holds.

On $\operatorname{Re} \left({s}\right) > 1$:

This last is $\zeta \left({2}\right)$ where $\zeta$ is the Riemann zeta function, so is finite by Analytic Continuation of Riemann Zeta Function.

Lemma 2
Define:


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

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

Let $G = \left({\Z / q \Z}\right)^\times$, and let $G^*$ be the dual group of characters on $G$.

Then for all $n \in \N$:


 * $\displaystyle \eta_{a, q} \left({n}\right) = \sum_{\chi \mathop \in G^*} \frac {\overline \chi \left({a}\right)} {\phi \left({q}\right)} \chi \left({n}\right)$

Proof of Lemma 2
There is only one $x \in G$ such that $\eta \left({x}\right) \ne 0$, and this equals $\eta \left({a}\right) = 1$.

So:


 * $\displaystyle \sum_{x \mathop \in G} \eta_{a, q} \left({x}\right) \overline \chi \left({x}\right) = \overline \chi \left({a}\right)$

Therefore, by Discrete Fourier Transform on Abelian Group we have for all $x \in G$:


 * $\displaystyle \eta \left({x}\right) = \frac 1 {\phi \left({q}\right)} \sum_{\chi \mathop \in G^*} \overline \chi \left({a}\right) \chi \left({x}\right)$

as required.

We have:

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

That is,:


 * $\displaystyle \sum_{p \mathop \in \mathcal P_{a, q} } \frac 1 {p^s} \sim \frac 1 {\phi \left({q}\right)} \ln \left(\frac 1 {s-1} \right)$

as $s \to 1$.