# 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

### 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.