View source for Dirichlet's theorem on arithmetic progressions

{{Short description|Theorem on the number of primes in arithmetic sequences}}
In [[number theory]], '''Dirichlet's theorem''', also called the Dirichlet [[prime number]] theorem, states that for any two positive [[coprime]] [[integer]]s ''a'' and&nbsp;''d'', there are infinitely many [[prime number|primes]] of the form ''a''&nbsp;+&nbsp;''nd'', where ''n'' is also a positive integer. In other words, there are infinitely many primes that are [[congruence relation|congruent]] to ''a'' [[modular arithmetic|modulo]] ''d''. The numbers of the form ''a''&nbsp;+&nbsp;''nd'' form an [[arithmetic progression]]

:<math>a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ </math>

and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem extends [[Euclid's theorem]] that there are infinitely many prime numbers (of the form 1 + 2n).  Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the [[multiplicative inverse|reciprocals]] of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among the congruence classes modulo ''d'' containing ''a'''s coprime to ''d''. 

The theorem is named after the German mathematician [[Peter Gustav Lejeune Dirichlet]], who proved it in 1837.

==Examples==
The primes of the form 4''n'' + 3 are {{OEIS|id=A002145}}
: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, ... 
They correspond to the following values of ''n'': {{OEIS|id=A095278}}
: 0, 1, 2, 4, 5, 7, 10, 11, 14, 16, 17, 19, 20, 25, 26, 31, 32, 34, 37, 40, 41, 44, 47, 49, 52, 55, 56, 59, 62, 65, 67, 70, 76, 77, 82, 86, 89, 91, 94, 95, ...
The strong form of Dirichlet's theorem implies that
:<math>\frac{1}{3}+\frac{1}{7}+\frac{1}{11}+\frac{1}{19}+\frac{1}{23}+\frac{1}{31}+\frac{1}{43}+\frac{1}{47}+\frac{1}{59}+\frac{1}{67}+\cdots</math>
is a [[divergent series]].

Sequences ''dn''&nbsp;+&nbsp;''a'' with odd ''d'' are often ignored because half the numbers are even and the other half is the same numbers as a sequence with 2''d'', if we start with ''n''&nbsp;=&nbsp;0. For example, 6''n''&nbsp;+&nbsp;1 produces the same primes as 3''n''&nbsp;+&nbsp;1, while 6''n''&nbsp;+&nbsp;5 produces the same as 3''n''&nbsp;+&nbsp;2 except for the only even prime 2. The following table lists several arithmetic progressions with infinitely many primes and the first few ones in each of them.
{| class = "wikitable"
|+
! Arithmetic<br>progression !! First 10 of infinitely many primes !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] sequence
|-
|2''n'' + 1 || 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … || {{OEIS link|id=A065091}}
|-
|4''n'' + 1 || 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, … || {{OEIS link|id=A002144}}
|-
|4''n'' + 3 || 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, … || {{OEIS link|id=A002145}}
|-
|6''n'' + 1 || 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, … || {{OEIS link|id=A002476}}
|-
|6''n'' + 5 || 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, … || {{OEIS link|id=A007528}}
|-
|8''n'' + 1 || 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, … || {{OEIS link|id=A007519}}
|-
|8''n'' + 3 || 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, … || {{OEIS link|id=A007520}}
|-
|8''n'' + 5 || 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, … || {{OEIS link|id=A007521}}
|-
|8''n'' + 7 || 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, … || {{OEIS link|id=A007522}}
|-
|10''n'' + 1 || 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, … || {{OEIS link|id=A030430}}
|-
|10''n'' + 3 || 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, … || {{OEIS link|id=A030431}}
|-
|10''n'' + 7 || 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, … || {{OEIS link|id=A030432}}
|-
|10''n'' + 9 || 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, … || {{OEIS link|id=A030433}}
|-
|12''n'' + 1 || 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, ... || {{OEIS link|id=A068228}}
|-
|12''n'' + 5 || 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, ... || {{OEIS link|id=A040117}}
|-
|12''n'' + 7 || 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, ... || {{OEIS link|id=A068229}}
|-
|12''n'' + 11 || 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, ... || {{OEIS link|id=A068231}}
|}
We can generate some forms of primes by using an iterative method. For example, we can generate primes of the form <math>4n+3</math> by using the following method:

Let <math>a_0=4(1)+3=7 </math>. Then we let <math>a_1=4a_0+3=4(7)+3=31</math> which is prime. We continue by computing <math>4(7)(31)+3=871=13(67)</math>. Because <math>4(7)(31)+3</math> is of the form <math>4n+3</math>, either 13 or 67 is of the form <math>4n+3</math>. We have that <math>67=4(16)+3</math>  and is prime, so <math>a_3=67</math>. We then continue this process to find successive primes of the form <math>4n+3</math> (Silverman 2013). 

==Distribution==
{{see also|Prime number theorem#Prime number theorem for arithmetic progressions}}
Since the primes thin out, on average, in accordance with the [[prime number theorem]], the same must be true for the primes in arithmetic progressions. It is natural to ask about the way the primes are shared between the various arithmetic progressions for a given value of ''d'' (there are ''d'' of those, essentially, if we do not distinguish two progressions sharing [[almost all]] their terms). The answer is given in this form: the number of feasible progressions ''modulo''&nbsp;''d'' — those where ''a'' and ''d'' do not have a common factor&nbsp;>&nbsp;1 — is given by [[Euler's totient function]]

:<math>\varphi(d).\ </math>

Further, the proportion of primes in each of those is

:<math>\frac {1}{\varphi(d)}.\ </math>

For example, if ''d'' is a prime number ''q'', each of the ''q''&nbsp;−&nbsp;1 progressions

:<math>q+1, 2 q+1, 3 q+1,\dots\ </math>
:<math>q+2, 2 q+2, 3 q+2,\dots\ </math>
:<math>\dots\ </math>
:<math>q+q-1, 2 q+q-1, 3 q+q-1,\dots\ </math>

(all except <math>q, 2q, 3q, \dots\ </math>)

contains a proportion 1/(''q''&nbsp;−&nbsp;1) of the primes.

When compared to each other, progressions with a quadratic nonresidue remainder have typically slightly more elements than those with a quadratic residue remainder ([[Chebyshev's bias]]).

==History==

In 1737, Euler related the study of prime numbers to what is known now as the Riemann zeta function: he showed that the value <math>\zeta(1)</math> reduces to a ratio of two infinite products, Π ''p'' / Π (''p''–1), for all primes ''p'', and that the ratio is infinite.<ref>{{cite journal |last= Euler |first= Leonhard |date= 1737 |title= Variae observationes circa series infinitas |trans-title= Various observations about infinite series |journal= Commentarii Academiae Scientiarum Imperialis Petropolitanae |volume= 9 |pages= 160–188 |url= https://books.google.com/books?id=cVo-AAAAcAAJ&pg=PA160 }}; specifically, [https://books.google.com/books?id=cVo-AAAAcAAJ&pg=PA172 Theorema 7 on pp. 172–174.]</ref><ref>Sandifer, C. Edward, ''The Early Mathematics of Leonhard Euler'' (Washington, D.C.: The Mathematical Association of America, 2007), [https://books.google.com/books?id=CvBxLr_0uBQC&pg=PA253 p.&nbsp;253.]</ref> In 1775, Euler stated the theorem for the cases of a + nd, where a = 1.<ref>Leonhard Euler, "De summa seriei ex numeris primis formatae 1/3 − 1/5 + 1/7 + 1/11 − 1/13 − 1/17 + 1/19 + 1/23 − 1/29 + 1/31 etc. ubi numeri primi formae 4''n'' − 1 habent signum positivum, formae autem 4''n'' + 1 signum negativum" (On the sum of series [composed] of prime numbers arranged 1/3 − 1/5 + 1/7 + 1/11 − 1/13 − 1/17 + 1/19 + 1/23 − 1/29 + 1/31 etc., where the prime numbers of the form 4''n'' − 1 have a positive sign, whereas [those] of the form 4''n'' + 1 [have] a negative sign.) in: Leonhard Euler, ''Opuscula analytica'' (St. Petersburg, Russia:  Imperial Academy of Sciences, 1785), vol. 2, pp.&nbsp;240–256; [https://books.google.com/books?id=spQ_AAAAcAAJ&pg=PA241 see p.&nbsp;241.]  From p.&nbsp;241:  ''"Quoniam porro numeri primi praeter binarium quasi a natura in duas classes distinguuntur, prouti fuerint vel formae 4n + 1, vel formae 4n − 1, dum priores omnes sunt summae duorum quadratorum, posteriores vero ab hac proprietate penitus excluduntur:  series reciprocae ex utraque classes formatae, scillicet:''
''1/5 + 1/13 + 1/17 + 1/29 + etc. et''
''1/3 + 1/7 + 1/11 + 1/19 + 1/23 + etc.''
''ambae erunt pariter infinitae, id quod etiam de omnibus speciebus numerorum primorum est tenendum.  Ita si ex numeris primis ii tantum excerpantur, qui sunt formae 100n + 1, cuiusmodi sunt 101, 401, 601, 701, etc., non solum multitudo eorum est infinita, sed etiam summa huius seriei ex illis formatae, scillicet:''
''1/101 + 1/401 + 1/601 + 1/701 + 1/1201 + 1/1301 + 1/1601 + 1/1801 + 1/1901 + etc.''
''etiam est infinita."'' (Since, further, prime numbers larger than two are divided as if by Nature into two classes, according as they were either of the form 4''n'' + 1, or of the form 4''n'' − 1, as all of the first are sums of two squares, but the latter are thoroughly excluded from this property:  reciprocal series formed from both classes, namely:
1/5 + 1/13 + 1/17 + 1/29 + etc. and
1/3 + 1/7 + 1/11 + 1/19 + 1/23 + etc.
will both be equally infinite, which [property] likewise is to be had from all types of prime numbers.  Thus, if there be chosen from the prime numbers only those that are of the form 100n + 1, of which kind are 101, 401, 601, 701, etc., not only the set of these is infinite, but likewise the sum of the series formed from that [set], namely:
1/101 + 1/401 + 1/601 + 1/701 + 1/1201 + 1/1301 + 1/1601 + 1/1801 + 1/1901 + etc.
likewise is infinite.)</ref> 
This special case of Dirichlet's theorem can be proven using [[cyclotomic polynomial]]s.{{sfnp|Neukirch|1999|loc=§I.10, Exercise 1}}
The general form of the theorem was first conjectured by [[Adrien-Marie Legendre|Legendre]] in his attempted unsuccessful proofs of [[quadratic reciprocity]]<ref>See:
*  Le Gendre (1785) [https://archive.org/stream/histoiredelacad85hist#page/464/mode/2up "Recherches d'analyse indéterminée"] (Investigations of interdeterminate analysis), ''Histoire de l'Académie royale des sciences, avec les mémoires de mathématique et de physique'', pp.&nbsp;465–559; [https://archive.org/stream/histoiredelacad85hist#page/552/mode/2up see especially p.&nbsp;552.]  From p.&nbsp;552:  ''"34.  ''Remarque''.  Il seroit peut-être nécessaire de démontrer rigoureusement une chose que nous avons supposée dans plusieurs endroits de cet article, savoir, qu'il y a une infinité de nombres premiers compris dans tous progression arithmétique, dont le premier terme & la raison sont premiers entr'eux, ou, ce qui revient au même, dans la formule 2mx + μ, lorsque 2m & μ n'ont point de commun diviseur.  Cette proposition est assez difficile à démontrer, cependant on peut s'assurer qu'elle est vraie, en comparant la progression arithmétique dont il s'agit, à la progression ordinaire 1, 3, 5, 7, &c.  Si on prend un grand nombre de termes de ces progressions, le même dans les deux, & qu'on les dispose, par exemple, de manière que le plus grand terme soit égal & à la même place de part & d'autre; on verra qu'en omettant de chaque côté les multiples de 3, 5, 7, &c. jusqu'à un certain nombre premier ''p'', il doit rester des deux côtés le même nombre de termes, ou même il en restera moins dans la progression 1, 3, 5, 7, &c.  Mais comme dans celle-ci, il reste nécessairement des nombres premiers, il en doit rester aussi dans l'autre."''  (34. ''Remark''.  It will perhaps be necessary to prove rigorously something that we have assumed at several places in this article, namely, that there is an infinitude of prime numbers included in every arithmetic progression, whose first term and common difference are co-prime, or, what amounts to the same thing, in the formula 2mx + μ, when 2m and μ have no common divisors at all.  This proposition is rather difficult to prove, however one may be assured that it is true, by comparing the arithmetic progression being considered to the ordinary progression 1, 3, 5, 7, etc.  If one takes a great number of terms of these progressions, the same [number of terms] in both, and if one arranges them, for example, in a way that the largest term be equal and at the same place in both; one will see that by omitting from each the multiples of 3, 5, 7, etc., up to a certain prime number ''p'', there should remain in both the same number of terms, or even there will remain fewer of them in the progression 1, 3, 5, 7, etc.  But as in this [set], there necessarily remain prime numbers, there shall also remain some in the other [set].)
*  A. M. Legendre, ''Essai sur la Théorie des Nombres'' (Paris, France:  Duprat, 1798), Introduction, [https://archive.org/stream/essaisurlathor00lege#page/8/mode/2up pp.&nbsp;9–16.]  From p.&nbsp;12:  ''"XIX. … En général, a étant un nombre donné quelconque, tout nombres impair peut être représenté par la formule 4ax ± b, dans laquelle b est impair et moindre que 2a.  Si parmi tous les valeurs possibles de b on retranche celles qui ont un commun diviseur avec a, les formes restantes 4ax ± b comprendront tous les nombres premiers partagé, … "''  (XIX. … In general, ''a'' being any given number, all odd numbers can be represented by the formula ''4ax ± b'', in which ''b'' is odd and less than ''2a''.  If among all possible values of ''b'' one removes those that have a common divisor with ''a'', the remaining formulas ''4ax ± b'' include all prime numbers among them … )
*  A. M. Legendre, ''Essai sur la Théorie des Nombres'', 2nd ed. (Paris, France:  Courcier, 1808), [https://books.google.com/books?id=ZHYAAAAAMAAJ&pg=PA404 p.&nbsp;404.]  From p.&nbsp;404:  ''"Soit donnée une progression arithmétique quelconque A − C, 2A − C, 3A − C, etc., dans laquelle A et C sont premiers entre eux; soit donnée aussi une suite θ, λ, μ … ψ, ω, composée de k nombres premiers impairs, pris à volonté et disposés dans un order quelconque; si on appelle en général π<sup>(z)</sup> le z<sup>ième</sup> terme de la suite naturelle des nombres premiers 3, 5, 7, 11, etc., je dis que sur π<sup>(k-1)</sup> termes consécutifs de la progression proposée, il y en aura au moins un qui ne sera divisible par aucun des nombres premiers θ, λ, μ … ψ, ω."''  (Let there be given any arithmetic progression ''A'' − ''C'', 2''A'' − ''C'', 3''A'' − ''C'', etc., in which ''A'' and ''C'' are prime among themselves [i.e., coprime]; let there be given also a series θ, λ, μ … ψ, ω composed of ''k'' odd prime numbers, taken at will and arranged in any order; if one calls in general π<sup>(''z'')</sup> the ''z''<sup>th</sup> term of the natural series of prime numbers 3, 5, 7, 11, etc., I claim that among the π<sup>(''k''−1)</sup> consecutive terms of the proposed progression, there will be at least one of them that will not be divisible by any of the prime numbers θ, λ, μ … ψ, ω.)  This assertion was proven false in 1858 by Anthanase Louis Dupré (1808–1869).  See:
:*  Dupré, A. (1859) [https://books.google.com/books?id=QH5aAAAAcAAJ&pg=PR3 ''Examen d'une proposition de Legendre relative à la théorie des nombres''] [Examination of a proposition of Legendre regarding the theory of numbers] (Paris, France:  Mallet-Bachelier, 1859).
:*   Narkiewicz, Władysław, ''The Development of Prime Number Theory: From Euclid to Hardy and Littlewood'' (Berlin, Germany:  Springer, 2000); [https://books.google.com/books?id=2gf9CAAAQBAJ&pg=PA50 see especially p.&nbsp;50.]</ref> — as [[Carl Friedrich Gauss|Gauss]] noted in his ''[[Disquisitiones Arithmeticae]]''<ref>Carl Friedrich Gauss, ''Disquisitiones arithmeticae'' (Leipzig, (Germany):  Gerhard Fleischer, Jr., 1801), Section 297, [https://babel.hathitrust.org/cgi/pt?id=nyp.33433070725894;view=1up;seq=527 pp. 507–508.]  From pp. 507–508:  ''"Ill. Le Gendre ipse fatetur, demonstrationem theorematis, sub tali forma ''kt'' + ''l'', designantibus ''k'', ''l'' numeros inter se primos datos, ''t'' indefinitum, certo contineri numeros primos, satis difficilem videri, methodumque obiter addigitat, quae forsan illuc conducere possit; multae vero disquisitiones praeliminares necessariae nobis videntur, antequam hacce quidem via ad demonstrationem rigorosam pervenire liceat."''  (The illustrious Le Gendre himself admits [that] the proof of the theorem — [namely, that] among [integers of] the form ''kt'' + ''l'', [where] ''k'' and ''l'' denote given integers [that are] prime among themselves [i.e., coprime] [and] ''t'' denotes a variable, surely prime numbers are contained — seems difficult enough, and incidentally, he points out a method that could perhaps lead to it; however, many preliminary and necessary investigations are [fore]seen by us before this [conjecture] may indeed reach the path to a rigorous proof.)</ref> — but it was proved by {{harvs|authorlink=Peter Gustav Lejeune Dirichlet|last=Dirichlet|year=1837|txt}} with [[Dirichlet L-series|Dirichlet ''L''-series]]. The proof is modeled on Euler's earlier work relating the [[Riemann zeta function]] to the distribution of primes. The theorem represents the beginning of rigorous [[analytic number theory]].

{{harvs|authorlink=Atle Selberg|first=Atle|last=Selberg|txt|year=1949}} gave an [[elementary proof]].

==Proof==
Dirichlet's theorem is proved by showing that the value of the [[Dirichlet L-function]] (of a non-trivial [[Dirichlet character|character]]) at 1 is nonzero. The proof of this statement requires some calculus and [[analytic number theory]] {{harv|Serre|1973}}. The particular case ''a'' = 1 (i.e., concerning the primes that are congruent to 1 modulo some ''n'') can be proven by analyzing the splitting behavior of primes in cyclotomic extensions, without making use of calculus {{harv|Neukirch|1999|loc=§VII.6}}.

Although the proof of Dirichlet's Theorem makes use of calculus and analytic number theory, some proofs of examples are much more straightforward. In particular, the proof of the example of infinitely many primes of the form <math>4n+3
</math> makes an argument similar to the one made in the proof of [[Euclid's theorem]] (Silverman 2013). The proof is given below:

We want to prove that there are infinitely many primes of the form <math>4n+3
</math>. Assume, for contradiction, that there are only finitely many primes of the form <math>4n+3
</math>. We then compile a list of all such primes <math>3,p_1,p_2,...,p_m
</math> where <math>p_1<p_2<...<p_m 
</math>. Let <math>N=4p_1p_2...p_m+3
</math>. It is clear that none of the primes in the list <math>3,p_1,p_2,...,p_m
</math> divide <math>N
</math>. Next, suppose that <math>N
</math> is composite. Then <math>N
</math> has unique prime factorization <math>N=a_1a_2...a_r 
</math> where each <math>a_i 
</math> is prime. Because <math>N\equiv 3\bmod 4
</math>, <math>N
</math> is odd and must be the product of only odd primes. Any odd prime <math>p
</math> must be such that <math>p\equiv 1\bmod 4
</math> or <math>p\equiv 3\bmod 4
</math>. It cannot be that <math>a_i\equiv 1\bmod 4 
</math> <math>\forall a_i 
</math> because if this were the case, then <math>N\equiv 1\bmod 4
</math>. So there exists a prime <math>a_i\equiv 3\bmod 4
</math> such that <math>a_i\mid N
</math>. However, none of the primes in the list <math>3,p_1,p_2,...,p_m
</math> divide <math>N
</math>, a contradiction. Therefore <math>N
</math> must be prime and <math>N>p_m
</math>. Hence, <math>N 
</math> is a prime of the form <math>4n+3
</math>, but it isn't included in the list <math>3,p_1,p_2,...,p_m
</math>. Thus, the list doesn't contain all such primes and there must be infinitely many primes of the form <math>4n+3
</math> (Silverman 2013).

==Generalizations==

The [[Bunyakovsky conjecture]] generalizes Dirichlet's theorem to higher-degree polynomials. Whether or not even simple quadratic polynomials such as {{nowrap|''x''<sup>2</sup> + 1}} (known from [[Landau's problems|Landau's fourth problem]]) attain infinitely many prime values is an important [[open problem]].

[[Dickson's conjecture]] generalizes Dirichlet's theorem to more than one polynomial.

[[Schinzel's hypothesis H]] generalizes these two conjectures, i.e. generalizes to more than one polynomial with degree larger than one.

In [[algebraic number theory]], Dirichlet's theorem generalizes to the [[Chebotarev's density theorem]].

[[Linnik's theorem]] (1944) concerns the size of the smallest prime in a given arithmetic progression. Linnik proved that the progression ''a''&nbsp;+&nbsp;''nd'' (as ''n'' ranges through the positive integers) contains a prime of magnitude at most ''cd<sup>L</sup>'' for absolute constants ''c'' and ''L''. Subsequent researchers have reduced ''L'' to 5.

An analogue of Dirichlet's theorem holds in the framework of dynamical systems ([[Toshikazu Sunada|T. Sunada]] and A. Katsuda, 1990).

Shiu showed that any arithmetic progression satisfying the hypothesis of Dirichlet's theorem will in fact contain arbitrarily long runs of ''consecutive'' prime numbers.<ref>{{cite journal | last1=Shiu | first1=D. K. L. | title=Strings of congruent primes | year=2000 | journal= J. London Math. Soc.| volume=61 |issue=2 | pages=359–373 | doi=10.1112/s0024610799007863}}</ref>

==See also==
*[[Bombieri–Vinogradov theorem]]
*[[Brun–Titchmarsh theorem]]
*[[Siegel–Walfisz theorem]]
*[[Dirichlet's approximation theorem]]
*[[Green–Tao theorem]]

==Notes==
{{reflist}}

==References==
*{{Apostol IANT}}
*{{MathWorld|title=Dirichlet's Theorem|urlname=DirichletsTheorem}}
*Chris Caldwell, [http://primes.utm.edu/notes/Dirichlet.html "Dirichlet's Theorem on Primes in Arithmetic Progressions"] at the [[Prime Pages]].
* {{Citation
| last=Dirichlet
| first=P. G. L.
| author-link=Peter Gustav Lejeune Dirichlet
| title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält
|trans-title=Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers
| journal=Abhandlungen der Königlichen Preußischen Akademie der Wissenschaften zu Berlin
| volume=48
| year=1837
| pages= 45–71
| url= https://babel.hathitrust.org/cgi/pt?id=chi.56785536;view=1up;seq=305
}}
*{{citation
 | last = Neukirch | first = Jürgen | author-link = Jürgen Neukirch
 | isbn = 3-540-65399-6
 | location = Berlin
 | mr = 1697859 | zbl=0956.11021
 | publisher = [[Springer-Verlag]]
 | series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
 | title = Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher
 | volume = 322
 | year = 1999}}.
*{{citation | doi=10.2307/1969454 | author-link=Atle Selberg | first=Atle | last=Selberg  | title=An elementary proof of Dirichlet's theorem about primes in an arithmetic progression | jstor=1969454 | journal=[[Annals of Mathematics]] | volume=50 | issue=2 | year=1949 | pages=297–304 | zbl=0036.30603 }}.
*{{citation | last=Serre  | first=Jean-Pierre | title=A course in arithmetic | zbl=0256.12001 | series=[[Graduate Texts in Mathematics]] | volume=7 | location=New York; Heidelberg; Berlin | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-90040-3 }}.
*{{citation | author-link=Toshikazu Sunada | first1=Toshikazu | last1=Sunada | first2=Atsushi | last2=Katsuda | title=Closed orbits in homology classes | journal=Publ. Math. IHÉS | volume=71 | year=1990 | pages=5–32 | doi=10.1007/BF02699875 | s2cid=26251216 | url=http://www.numdam.org/item/PMIHES_1990__71__5_0/ }}.
*[https://www.pearson.com/en-us/pearsonplus/p/9780137981175 Silverman JH (2013) A Friendly Introduction to Number Theory: Pearson New International Edition, Pearson Education.]

==External links==
* [https://bibliothek.bbaw.de/digitalisierte-sammlungen/akademieschriften/ansicht-akademieschriften?tx_bbaw_academicpublicationshow%5baction%5d=show&tx_bbaw_academicpublicationshow%5bcontroller%5d=AcademicPublication\Volume&tx_bbaw_academicpublicationshow%5bpage%5d=286&tx_bbaw_academicpublicationshow%5bvolume%5d=108&cHash=cdcc91473488a5ab2956096880af8f0f Scans of the original paper in German]
* [https://arxiv.org/abs/0808.1408 Dirichlet: ''There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime''] English translation of the original paper at the arXiv
* [http://demonstrations.wolfram.com/DirichletsTheorem/ Dirichlet's Theorem] by Jay Warendorff, [[Wolfram Demonstrations Project]].

{{Peter Gustav Lejeune Dirichlet}}

[[Category:Theorems about prime numbers]]
[[Category:Zeta and L-functions]]