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 a non-negative integer. In other words, there are infinitely many primes which are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression
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 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.
Dirichlet's theorem does not require that the sequence contains only prime numbers and deals with infinite sequences. For finite sequences, there exist arbitrarily long arithmetic progressions of primes, a theorem known as the Green–Tao theorem.
An integer is a prime for the Gaussian integers if either the square of its modulus is a prime number (in the normal sense) or one of its parts is zero and the absolute value of the other is a prime that is congruent to 3 modulo 4. The primes (in the normal sense) of the type 4n + 3 are (sequence A002145 in the OEIS)
- 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, ...
- 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
is a divergent series.
The following table lists several arithmetic progressions with infinitely many primes and the first few ones in each of them.
|First 10 of infinitely many primes||OEIS sequence|
|2n + 1||3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …||A065091|
|4n + 1||5, 13, 17, 29, 37, 41, 53, 61, 73, 89, …||A002144|
|4n + 3||3, 7, 11, 19, 23, 31, 43, 47, 59, 67, …||A002145|
|6n + 1||7, 13, 19, 31, 37, 43, 61, 67, 73, 79, …||A002476|
|6n + 5||5, 11, 17, 23, 29, 41, 47, 53, 59, 71, …||A007528|
|8n + 1||17, 41, 73, 89, 97, 113, 137, 193, 233, 241, …||A007519|
|8n + 3||3, 11, 19, 43, 59, 67, 83, 107, 131, 139, …||A007520|
|8n + 5||5, 13, 29, 37, 53, 61, 101, 109, 149, 157, …||A007521|
|8n + 7||7, 23, 31, 47, 71, 79, 103, 127, 151, 167, …||A007522|
|10n + 1||11, 31, 41, 61, 71, 101, 131, 151, 181, 191, …||A030430|
|10n + 3||3, 13, 23, 43, 53, 73, 83, 103, 113, 163, …||A030431|
|10n + 7||7, 17, 37, 47, 67, 97, 107, 127, 137, 157, …||A030432|
|10n + 9||19, 29, 59, 79, 89, 109, 139, 149, 179, 199, …||A030433|
|12n + 1||13, 37, 61, 73, 97, 109, 157, 181, 193, 229, ...||A068228|
|12n + 5||5, 17, 29, 41, 53, 89, 101, 113, 137, 149, ...||A040117|
|12n + 7||7, 19, 31, 43, 67, 79, 103, 127, 139, 151, ...||A068229|
|12n + 11||11, 23, 47, 59, 71, 83, 107, 131, 167, 179, ...||A068231|
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
Further, the proportion of primes in each of those is
For example if d is a prime number q, each of the q − 1 progressions, other than
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).
Some preliminary work on the problem was done by the Swiss mathematician Leonhard Euler in 1737. Specifically, he proved the sum-product formula for the Riemann zeta function for the value s=1. However, he did not conclude from that result that there was an infinitude of primes. This special case has a short elementary proof using cyclotomic polynomials. In 1785, Euler stated the theorem for the cases of a + nd, where a = 1. The general form of the theorem was first conjectured by Legendre in his attempted unsuccessful proofs of quadratic reciprocity — as Gauss noted in his Disquisitiones Arithmeticae — but it was proved by Dirichlet (1837) 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.
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 (Serre 1973). 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 (Neukirch 1999).
The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-order 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.
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).
- Bombieri–Vinogradov theorem
- Brun–Titchmarsh theorem
- Siegel–Walfisz theorem
- Dirichlet's approximation theorem
- Euler, Leonhard (1737). "Variae observationes circa series infinitas" [Various observations about infinite series]. Commentarii academiae scientiarum imperialis Petropolitanae 9: 160–188. ; specifically, Theorema 7 on pp. 172–174.
- Euler's article is discussed in: C. Edward Sandifer, The Early Mathematics of Leonhard Euler (Providence, Rhode Island: The Mathematical Society of America, 2007), pp. 249–260. On p. 254, Sandifer states: "This result [i.e., Theorem 7] is sometimes cited as Euler's proof that there are infinitely many prime numbers. Because the harmonic series diverges, the product diverges, and a product can diverge only if it is an infinite product. Hence there must be infinitely many prime numbers. Though the infinitude of primes is immediate, Euler does not explicitly make the connection here."
- Leonard 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: Leonard 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.)
- 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 which 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.
- Carl Friedrich Gauss, Disquisitiones arithmeticae (Leipzig, (Gemany): Gerhard Fleischer, Jr., 1801), Section 297, 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, denoting k and l [as] given integers [that are] prime among themselves [i.e., coprime] [and] t [as 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.)
- Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
- Weisstein, Eric W., "Dirichlet's Theorem", MathWorld.
- Chris Caldwell, "Dirichlet's Theorem on Primes in Arithmetic Progressions" at the Prime Pages.
- Dirichlet, P. G. L. (1837), "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält" [Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contain infinitely many prime numbers], Abhandlungen der Königlichen Preußischen Akademie der Wissenschaften zu Berlin 48: 45–71
- Neukirch, Jürgen (1999), Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 322, Berlin: Springer-Verlag, Section VII.6 and Exercise I.10.1, ISBN 3-540-65399-6, MR 1697859, Zbl 0956.11021.
- Selberg, Atle (1949), "An elementary proof of Dirichlet's theorem about primes in an arithmetic progression", Annals of Mathematics 50 (2): 297–304, doi:10.2307/1969454, JSTOR 1969454, Zbl 0036.30603.
- Serre, Jean-Pierre (1973), A course in arithmetic, Graduate Texts in Mathematics 7, New York-Heidelberg-Berlin: Springer- Verlag, ISBN 3-540-90040-3, Zbl 0256.12001.
- Sunada, Toshikazu; Katsuda, Atsushi (1990), "Closed orbits in homology classes", Publ. Math. IHES 71: 5–32.
- Scans of the original paper in German
- 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
- Dirichlet's Theorem by Jay Warendorff, Wolfram Demonstrations Project.