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:

$p \equiv a (\mod D)$ .

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

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 $O (D^{1 + ϵ})$ .
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 $O (D^{2 + ϵ})$ .
Linnick'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 $L = 5.5$ .

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.