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 $p \equiv a \text{ mod } q$.

Then $\mathcal P_{a,q}$ has Dirichlet density $\phi(q)^{-1}$, 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 $\displaystyle f(s) =\sum_p \chi(p) p^{-s}$

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

If $\chi$ is the trivial character then


 * $\displaystyle f(s) \sim \log \left( \frac1{s-1} \right)$

as $s \to 1$.

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


 * $\displaystyle \sum_p\chi(p) p^{-s} = \log L(s,\chi) - \sum_p \sum_{n \geq 2} \frac{\chi(p)^n}{n p^{ns}} \qquad (1)$

If $\chi$ is non-trivial, then by L-functions Do Not Vanish at One, $\log L(s,\chi)$ is bounded as $s \to 1$.

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

Therefore in this case, $\displaystyle L(s,\chi) \sim \frac\lambda{s-1}$ where $\lambda$ is the residue of $L(s,\chi)$ at $1$, and


 * $\displaystyle \log L(s,\chi) \sim \log\left( \frac\lambda{s-1} \right) \sim \log\left( \frac1{s-1} \right)$

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

On $\Re(s) > 1$,

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

Lemma 2
Define


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

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

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

Then for all $n \in \N$,


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

Proof of Lemma 2
There is only one $x \in G$ such that $\eta(x) \neq 0$, and this equals $\eta(a) = 1$, so


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

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


 * $\displaystyle \eta(x) = \frac1{\phi(q)}\sum_{\chi \in G^*} \overline{\chi}(a) \chi(x)$

as required.

We have

By Lemma 1, the first term grows like $\displaystyle \frac1{\phi(q)}\log \frac1{s-1}$ as $s \to 1$, while all other terms are bounded. That is,


 * $\displaystyle \sum_{p \in \mathcal P_{a,q}} \frac 1{p^s} \sim \frac 1{\phi(q)} \log\left(\frac1{s-1} \right)$

as $s \to 1$.