Dirichlet's theorem on arithmetic progressions

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression


 * $$a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ $$

and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, 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 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.

Examples
The primes of the form 4n + 3 are
 * 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:
 * 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
 * $$\frac{1}{3}+\frac{1}{7}+\frac{1}{11}+\frac{1}{19}+\frac{1}{23}+\frac{1}{31}+\frac{1}{43}+\frac{1}{47}+\frac{1}{59}+\frac{1}{67}+\cdots$$

is a divergent series.

Sequences dn + 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 2d, if we start with n = 0. For example, 6n + 1 produces the same primes as 3n + 1, while 6n + 5 produces the same as 3n + 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.

Distribution
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 d — those where a and d do not have a common factor > 1 — is given by Euler's totient function


 * $$\varphi(d).\ $$

Further, the proportion of primes in each of those is


 * $$\frac {1}{\varphi(d)}.\ $$

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


 * $$q+1, 2 q+1, 3 q+1,\dots\ $$
 * $$q+2, 2 q+2, 3 q+2,\dots\ $$
 * $$\dots\ $$
 * $$q+q-1, 2 q+q-1, 3 q+q-1,\dots\ $$

(all except $$q, 2q, 3q, \dots\ $$)

contains a proportion 1/(q − 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 $$\zeta(1)$$ reduces to a ratio of two infinite products, Π p / Π (p–1), for all primes p, and that the ratio is infinite. In 1775, Euler stated the theorem for the cases of a + nd, where a = 1. 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 4n − 1 habent signum positivum, formae autem 4n + 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 4n − 1 have a positive sign, whereas [those] of the form 4n + 1 [have] a negative sign.) in: Leonhard Euler, Opuscula analytica (St. Petersburg, Russia: Imperial Academy of Sciences, 1785), vol. 2, pp. 240–256; see p. 241.  From p. 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 4n + 1, or of the form 4n − 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.) This special case of Dirichlet's theorem can be proven using cyclotomic polynomials. The general form of the theorem was first conjectured by Legendre in his attempted unsuccessful proofs of quadratic reciprocity See:
 * Le Gendre (1785) "Recherches d'analyse indéterminée" (Investigations of interdeterminate analysis), Histoire de l'Académie royale des sciences, avec les mémoires de mathématique et de physique, pp. 465–559; see especially p. 552. From p. 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, pp. 9–16.  From p. 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), p. 404.  From p. 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(z) le zième terme de la suite naturelle des nombres premiers 3, 5, 7, 11, etc., je dis que sur π(k-1) 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, 2A − C, 3A − 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(z) the zth term of the natural series of prime numbers 3, 5, 7, 11, etc., I claim that among the π(k−1) 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) 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); see especially p. 50. — as Gauss noted in his Disquisitiones Arithmeticae — but it was proved by  with 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.

gave an elementary proof.

Proof
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. 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.

Generalizations
The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-degree polynomials. Whether or not even simple quadratic polynomials such as x2 + 1 (known from Landau's fourth problem) attain infinitely many prime values is an important open problem.

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

The 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 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 + nd (as n ranges through the positive integers) contains a prime of magnitude at most cdL 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 (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.