# 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

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, ...

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

${\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{19}}+{\frac {1}{23}}+{\frac {1}{31}}+{\frac {1}{43}}+{\frac {1}{47}}+{\frac {1}{59}}+{\frac {1}{67}}+\cdots$ is a divergent series.

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 primesOEIS sequence
2n + 13, 5, 7, 11, 13, 17, 19, 23, 29, 31, …A065091
4n + 15, 13, 17, 29, 37, 41, 53, 61, 73, 89, …A002144
4n + 33, 7, 11, 19, 23, 31, 43, 47, 59, 67, …A002145
6n + 17, 13, 19, 31, 37, 43, 61, 67, 73, 79, …A002476
6n + 55, 11, 17, 23, 29, 41, 47, 53, 59, 71, …A007528
8n + 117, 41, 73, 89, 97, 113, 137, 193, 233, 241, …A007519
8n + 33, 11, 19, 43, 59, 67, 83, 107, 131, 139, …A007520
8n + 55, 13, 29, 37, 53, 61, 101, 109, 149, 157, …A007521
8n + 77, 23, 31, 47, 71, 79, 103, 127, 151, 167, …A007522
10n + 111, 31, 41, 61, 71, 101, 131, 151, 181, 191, …A030430
10n + 33, 13, 23, 43, 53, 73, 83, 103, 113, 163, …A030431
10n + 77, 17, 37, 47, 67, 97, 107, 127, 137, 157, …A030432
10n + 919, 29, 59, 79, 89, 109, 139, 149, 179, 199, …A030433
12n + 113, 37, 61, 73, 97, 109, 157, 181, 193, 229, ...A068228
12n + 55, 17, 29, 41, 53, 89, 101, 113, 137, 149, ...A040117
12n + 77, 19, 31, 43, 67, 79, 103, 127, 139, 151, ...A068229
12n + 1111, 23, 47, 59, 71, 83, 107, 131, 167, 179, ...A068231

## 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, other than

$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. 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 — 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.

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) 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).

## 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).

## See also

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