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:

$\map \phi q^{-1}$

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:

$\displaystyle \map f s = \sum_p \map \chi p p^{-s}$

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

If $\chi$ is the trivial character then:

$\map f s \sim \map \ln {\dfrac 1 {s - 1} }$

as $s \to 1$.

$\Box$


Define:

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


Lemma 2

Let $G = \paren {\Z / q \Z}^\times$.

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



Then for all $n \in \N$:

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

$\Box$


We have:

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

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 $s \to 1$.

$\blacksquare$


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