Dirichlet's Theorem on Arithmetic Progressions

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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.


Proof

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:

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

as $s \to 1$.

$\Box$


Lemma 2

Define:

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

Let $G = \left({\Z / q \Z}\right)^\times$.

Let $G^*$ be the dual group of characters on $G$.



Then for all $n \in \N$:

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

$\Box$


We have:

\(\displaystyle \sum_{p \mathop \in \mathcal P_{a, q} } p^{-s}\) \(=\) \(\displaystyle \sum_p \eta_{a, q} \left({p}\right) p^{-s}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_p \sum_{\chi \mathop \in G^*} \frac {\overline \chi \left({a}\right)} {\phi \left({q}\right)} \chi \left({p}\right) p^{-s}\) $\quad$ Lemma 2 $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\phi \left({q}\right)} \sum_p \frac {\chi_0 \left({p}\right)} {p^s} + \sum_{\substack {\chi \mathop \in G^* \\ \chi \mathop \ne \chi_0} } \frac {\overline \chi \left({a}\right)} {\phi \left({q}\right)} \sum_p \chi \left({p}\right) p^{-s}\) $\quad$ where $\chi_0$ is the trivial character on $G$ $\quad$

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,:

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

as $s \to 1$.

$\blacksquare$


Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.


Historical Note

Dirichlet's Theorem on Arithmetic Progressions was first proved by Peter Dirichlet in $1837$.


Sources