Dirichlet's Theorem on Arithmetic Sequences

From ProofWiki
Jump to navigation Jump to search

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 $G$.


This article, or a section of it, needs explaining.
In particular: Explain the notation $\paren {\Z / q \Z}^\times$
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


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 $s \to 1$.

$\blacksquare$


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

Retrieved from "https://proofwiki.org/w/index.php?title=Dirichlet%27s_Theorem_on_Arithmetic_Sequences&oldid=529041"