Dirichlet's theorem on primes in arithmetic progressions

Statement

Let be relatively prime natural numbers. Then, there exist infinitely many primes such that:

.

For fixed , the primes that are congruent to modulo are termed Dirichlet primes.

Related facts

Stronger facts

• Chebotarev density theorem

Easy cases

• There are infinitely many primes that are one modulo any modulus
• Dirichlet's theorem for modulus four
• Dirichlet's theorem for modulus eight

Related facts about infinitude

• Infinitude of primes
• Set of primes is large

Conjectures/facts about the first Dirichlet prime

• Heath-Brown's conjecture on the first Dirichlet prime: A conjecture, saying that the first Dirichlet prime in a given congruence class modulo is .
• Chowla's conjecture on the first Dirichlet prime: A conjecture, saying that the first Dirichlet prime in a given congruence class modulo is .
• Chowla's corollary to generalized Riemannn hypothesis: Proved conditional to the generalized Riemann hypothesis, saying that the first Dirichlet prime in a given congruence class modulo is .
• Linnik's theorem: An unconditional theorem, saying that there exists such that the first Dirichlet prime in a given congruence class modulo is . Heath-Brown showed that we can take .

Conjectures/facts about contiguous blocks of Dirichlet primes

• Green-Tao theorem: This states that for any , there exists a prime arithmetic progression of length : an arithmetic progression of length , all of whose members are primes.

Conjectures that generalize to higher degree polynomials

• Bunyakovsky conjecture is a conjecture for polynomials of degree two or more whose analogue for linear polynomials would be Dirichlet's theorem.