Definitions

# 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 ≥ 0, or in other words: there are infinitely many primes which are congruent to a modulo d. Moreover, the sum of the reciprocals of such primes diverges.

This theorem extends Euclid's theorem that there are infinitely many primes (in this case of the form 3 + 4n, which are also the Gaussian primes, or of the form 1 + 2n, for every odd number, excluding 1). Note that the theorem does not say that there are infinitely many consecutive terms in the arithmetic progression

a, a+d, a+2d, a+3d, ...,

which are prime.

## Examples

We get primes of the type 4n + 3 'only' for n with the values
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, ….

Arithmetic
progression
First 10 of infinitely many primes OEIS id
2n + 1 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
4n + 1 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, …
4n + 3 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, …
6n + 1 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, …
6n + 5 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, …
8n + 1 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, …
8n + 3 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, …
8n + 5 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, …
8n + 7 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, …
10n + 1 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, …
10n + 3 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, …
10n + 7 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, …
10n + 9 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, …

## Distribution

Since the primes thin out, on average, 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

φ(d).

Further, the proportion of primes in each of those is

1/φ(d).

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

q, 2q, 3q, ...

contains a proportion 1/(q − 1) of the primes.

## History

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 Gauss and proved by Dirichlet in 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.

In algebraic number theory Dirichlet's theorem generalizes to Chebotarev's density theorem.