Heath-Brown's conjecture on the first Dirichlet prime

Statement

Quick statmeent

The first Dirichlet prime in any relatively prime congruence class modulo $D$ is $O(D(\log D{)}^2)$.

Statement in terms of the first Dirichlet prime

There exists a constant $C$ such that the following holds:

For any relatively prime natural numbers $a,D$, there exists a prime $p$ such that $p\equiv a(\mathrm{mod}D)$ and $p<CD(\log D{)}^2$.

Relation with other conjectures and facts

Weaker conjectures and facts

• Chowla's conjecture on the first Dirichlet prime gives a bound of $O(D^{1+ϵ})$.
• Chowla's corollary to generalized Riemann hypothesis gives a bound of $O(D^{2+ϵ})$, conditional to the generalized Riemann hypothesis.
• Linnik's theorem gives a bound of $O(D^L)$ for some large but finite $L$. Heath-Brown has shown that we can choose $L\le 5.5$.