Dirichlet's Theorem on Arithmetic Sequences

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Theorem

Let $a, q$ be coprime integers.

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


Then $\PP_{a, q}$ has Dirichlet density:

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

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


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


Proof

Lemma 1

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

Let:

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

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 Dirichlet characters on $G$.


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Then for all $n \in \N$:

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

$\Box$


We have:

\(\ds \sum_{p \mathop \in \PP_{a, q} } p^{-s}\) \(=\) \(\ds \sum_p \map {\eta_{a, q} } p p^{-s}\)
\(\ds \) \(=\) \(\ds \sum_p \sum_{\chi \mathop \in G^*} \frac {\map {\overline \chi} a} {\map \phi q} \, \map \chi p p^{-s}\) Lemma 2
\(\ds \) \(=\) \(\ds \frac 1 {\map \phi q} \sum_p \frac {\map {\chi_0} p} {p^s} + \sum_{\substack {\chi \mathop \in G^* \\ \chi \mathop \ne \chi_0} } \frac {\map {\overline \chi} a} {\map \phi q} \sum_p \map \chi p p^{-s}\) 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:

$\ds \sum_{p \mathop \in \PP_{a, q} } \frac 1 {p^s} \sim \frac 1 {\map \phi q} \, \map \ln {\dfrac 1 {s - 1} }$

as $s \to 1$.

$\blacksquare$


Also known as

Dirichlet's Theorem on Arithmetic Sequences is also known just as Dirichlet's Theorem.

However, there is more than one theorem named such, so it is preferable to use the full form.


Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.


Historical Note

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


Sources

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