Dirichlet's theorem on primes in arithmetic progressions

Statement

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

$p\equiv a{\pmod {D}}$.

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

Related facts

Stronger facts

• Chebotarev density theorem

Easy case

• 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 $D$ is $O(D(\log D)^{2})$.
• Chowla's conjecture on the first Dirichlet prime: A conjecture, saying that the first Dirichlet prime in a given congruence class modulo $D$ is $O(D^{1+\epsilon })$.
• 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 $D$ is $O(D^{2+\epsilon })$.
• Linnik's theorem: An unconditional theorem, saying that there exists $L>0$ such that the first Dirichlet prime in a given congruence class modulo $D$ is $O(D^{L})$. Heath-Brown showed that we can take $L=5.5$.

Conjectures/facts about contiguous blocks of Dirichlet primes

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