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 Failed to parse (unknown function "\pomd"): {\displaystyle p \equiv a \pomd 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 $O(D^{1+\epsilon })$.
• Chowla's corollary to generalized Riemann hypothesis gives a bound of $O(D^{2+\epsilon })$, conditional to the generalized Riemann hypothesis.
• Linnick's theorem gives a bound of $O(D^{L})$ for some large but finite $L$. Heath-Brown has shown that we can choose $L\leq 5.5$.