Dirichlet's Theorem on Arithmetic Sequences
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 $s \to 1$.
$\blacksquare$
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
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.2$: Divisibility and factorization in $\mathbf Z$: Exercise $9$ (mentioned, but not proved)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.28$: Dirichlet ($\text {1805}$ – $\text {1859}$)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes: Theorem $4$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Dirichlet's theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Dirichlet's theorem