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| In [[number theory]], '''Dirichlet's theorem''', also called the Dirichlet prime number theorem, states that for any two positive [[coprime]] [[integer]]s ''a'' and ''d'', there are infinitely many [[prime number|primes]] of the form ''a'' + ''nd'', where n is a non-negative integer. In other words, there are infinitely many primes which are [[congruence relation|congruent]] to ''a'' [[modular arithmetic|modulo]] ''d''. The numbers of the form ''a'' + ''nd'' form an [[arithmetic progression]]
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| :<math>a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ </math>
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| 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. 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''. | |
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| Note that Dirichlet's theorem does ''not'' require the prime numbers in an arithmetic sequence to be consecutive. It is also known that there exist arbitrarily long finite [[Primes in arithmetic progression|arithmetic progressions consisting only of primes]], but this is a different result, known as the [[Green–Tao theorem]].
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| ==Examples==
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| An integer is a prime for the [[Gaussian integer]]s [[If and only if|iff]] its absolute value is a prime number (in the normal sense) that is congruent to 3 modulo 4.
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| The primes (in the normal sense) of the type 4''n'' + 3 are
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| : 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, ….
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| They correspond to the following values of ''n'':
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| : 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, ….
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| The strong form of Dirichlet's theorem implies that
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| :<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>
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| is a [[divergent series]]. | |
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| The following table lists several arithmetic progressions with infinite primes and the first few ones in each of them.
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| {| class = "wikitable"
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| |+
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| ! Arithmetic<br>progression !! First 10 of infinitely many primes !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] sequence
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| |2''n'' + 1 || 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … || {{OEIS link|id=A065091}}
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| |-
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| |4''n'' + 1 || 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, … || {{OEIS link|id=A002144}}
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| |-
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| |4''n'' + 3 || 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, … || {{OEIS link|id=A002145}}
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| |6''n'' + 1 || 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, … || {{OEIS link|id=A002476}}
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| |6''n'' + 5 || 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, … || {{OEIS link|id=A007528}}
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| |8''n'' + 1 || 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, … || {{OEIS link|id=A007519}}
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| |8''n'' + 3 || 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, … || {{OEIS link|id=A007520}}
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| |8''n'' + 5 || 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, … || {{OEIS link|id=A007521}}
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| |8''n'' + 7 || 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, … || {{OEIS link|id=A007522}}
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| |10''n'' + 1 || 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, … || {{OEIS link|id=A030430}}
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| |10''n'' + 3 || 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, … || {{OEIS link|id=A030431}}
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| |10''n'' + 7 || 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, … || {{OEIS link|id=A030432}}
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| |10''n'' + 9 || 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, … || {{OEIS link|id=A030433}}
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| |}
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| ==Distribution==
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| 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. One naturally then asks 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 don't distinguish two progressions sharing [[almost all]] their terms). The answer is given in this form: the number of feasible progressions ''modulo'' ''d'' — those where ''a'' and ''d'' do not have a common factor > 1 — is given by [[Euler's totient function]]
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| :<math>\varphi(d).\ </math>
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| Further, the proportion of primes in each of those is
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| :<math>\frac {1}{\varphi(d)}.\ </math>
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| For example if ''d'' is a prime number ''q'', each of the ''q'' − 1 progressions, other than
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| :<math>q, 2q, 3q, \dots\ </math>
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| contains a proportion 1/(''q'' − 1) of the primes.
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| 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]]).
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| ==History==
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| [[Leonhard Euler|Euler]] stated that every arithmetic progression beginning with 1 contains an infinite number of primes. The theorem in the above form was first conjectured by [[Adrien-Marie Legendre|Legendre]] in his attempted unsuccessful proofs of [[quadratic reciprocity]] and 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]].
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| {{harvs|authorlink=Atle Selberg|first=Atle|last=Selberg|txt|year=1946}} gave an [[elementary proof]].
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| ==Proof==
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| Dirichlet's theorem is proved by showing that the value of the [[Dirichlet L-function]] (of a non-trivial character) at 1 is nonzero. The proof of this statement requires some calculus and [[analytic number theory]]. In 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}}.
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| ==Generalizations==
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| The [[Bunyakovsky conjecture]] generalizes Dirichlet's theorem to higher-order polynomials. Whether or not even simple quadratic polynomials such as {{nowrap|''x''<sup>2</sup> + 1}} attain infinitely many prime values is an important [[open problem]].
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| In [[algebraic number theory]], Dirichlet's theorem generalizes to [[Chebotarev's density theorem]].
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| [[Linnik's theorem]] (1944) concerns the size of the smallest prime in a given arithmetic progression. Linnik proved that the progression ''a'' + ''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 approximately 5.2.
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| ==See also==
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| *[[Bombieri–Vinogradov theorem]]
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| *[[Brun–Titchmarsh theorem]]
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| *[[Siegel–Walfisz theorem]]
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| *[[Dirichlet's approximation theorem]]
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| ==References==
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| *{{Apostol IANT}}
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| *{{MathWorld|title=Dirichlet's Theorem|urlname=DirichletsTheorem}}
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| *Chris Caldwell, [http://primes.utm.edu/notes/Dirichlet.html "Dirichlet's Theorem on Primes in Arithmetic Progressions"] at the [[Prime Pages]].
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| * {{Citation
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| | last=Dirichlet
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| | first=P. G. L.
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| | author-link=Peter Gustav Lejeune Dirichlet
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| | title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält
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| | journal=Abhand. Ak. Wiss. Berlin
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| | volume=48
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| | year=1837
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| }}
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| *{{citation
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| | last = Neukirch | first = Jürgen | authorlink = Jürgen Neukirch
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| | isbn = 3-540-65399-6
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| | location = Berlin
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| | mr = 1697859
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| | at = Section VII.6 and Exercise I.10.1
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| | publisher = Springer-Verlag
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| | series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
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| | title = Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher
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| | volume = 322
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| | year = 1999}}.
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| *{{citation|doi=10.2307/1969454|authorlink=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}}.
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| ==External links==
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| * [http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige?band=07-abh/1837&seite:int=00000286 Scans of the original paper in German]
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| * [http://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
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| * [http://demonstrations.wolfram.com/DirichletsTheorem/ Dirichlet's Theorem] by Jay Warendorff, [[Wolfram Demonstrations Project]].
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| [[Category:Theorems about prime numbers]]
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| [[Category:Zeta and L-functions]]
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