Chowla's conjecture on the first Dirichlet prime

Statement

In terms of the first Dirichlet prime

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

Suppose $a$ and $D$ are relatively prime natural numbers. Then, there exists a prime $p\equiv a{\pmod {D}}$ such that $p<CD^{1+\epsilon }$.

In terms of the first few Dirichlet primes

For any $\epsilon >0$ 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 $p\equiv a{\pmod {D}}$ such that $p<CD^{1+\epsilon }$.

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 $2+\epsilon$ instead of $1+\epsilon$.
• Linnick's theorem: This is an unconditional version where $1+\epsilon$ is replaced by some large constant $L$. Heath-Brown have shown that $L\leq 5.5$.