# Dirichlet's theorem on arithmetic progressions

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. Autrement dit, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression {style d'affichage a, a+d, a+2d, a+3d, des points , } and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. Le théorème, 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. De manière équivalente, the primes are evenly distributed (asymptotiquement) among the congruence classes modulo d containing a's coprime to d.

Contenu 1 Exemples 2 Distribution 3 Histoire 4 Preuve 5 Généralisations 6 Voir également 7 Remarques 8 Références 9 External links Examples The primes of the form 4n + 3 sommes (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, ...

They correspond to the following values of n: (sequence A095278 in the OEIS) 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 {style d'affichage {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. Par exemple, 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.

Arithmetic progression 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 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, essentiellement, 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 {style d'affichage varphi (ré). } Plus loin, the proportion of primes in each of those is {style d'affichage {frac {1}{varphi (ré)}}. } Par exemple, if d is a prime number q, each of the q − 1 progressions {displaystyle q+1,2q+1,3q+1dots } {displaystyle q+2,2q+2,3q+2dots } {displaystyle dots } {displaystyle q+q-1,2q+q-1,3q+q-1dots } (all except {style d'affichage q,2q,3q,des points } ) 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 {displaystyle 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.[1][2] Dans 1775, Euler stated the theorem for the cases of a + nd, where a = 1.[3] This special case of Dirichlet's theorem can be proven using cyclotomic polynomials.[4] The general form of the theorem was first conjectured by Legendre in his attempted unsuccessful proofs of quadratic reciprocity[5] — as Gauss noted in his Disquisitiones Arithmeticae[6] — 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.

Atle Selberg (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 character) à 1 is nonzero. The proof of this statement requires some calculus and analytic number theory (Serre 1973). In the particular case a = 1 (c'est à dire., 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, §VII.6).

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, c'est à dire. 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 (J. Sunada and A. Katsuda, 1990).