Chowla's conjecture on the first Dirichlet prime

Statement

Quick statement

The first Dirichlet prime in any relatively prime congruence class modulo $D$ is $O(D^{1+ϵ})$.

In terms of the first Dirichlet prime

For any $ϵ>0$, there exists a constant $C$ such that the following holds:

Suppose $a$ and $D$ are relatively prime natural numbers. Then, there exists a prime $p\equiv a(\mathrm{mod}D)$ such that $p<C{D}^{1+ϵ}$.

In terms of the first few Dirichlet primes

For any $ϵ>0$ and any natural number $k$, there exists a constant $C$ such that the following holds:

Suppose $a$ and $D$ are relatively prime natural numbers. Then, there exist at least $k$ distinct primes $p\equiv a(\mathrm{mod}D)$ such that $p<C{D}^{1+ϵ}$.

This follows from the version involving the first Dirichlet prime.

Relation with other facts

Stronger conjectures

• Heath-Brown's conjecture on the first Dirichlet prime

Weaker facts and conjectures

• Chowla's corollary to generalized Riemann hypothesis: Under the [generalized Riemann hypothesis]], we have the analogous result for $2+ϵ$ instead of $1+ϵ$.
• Linnick's theorem: This is an unconditional version where $1+ϵ$ is replaced by some large constant $L$. Heath-Brown have shown that $L\le 5.5$.