Linnik's Theorem

Let $p(d,a)$ be the smallest Prime in the arithmetic progression $\{a+kd\}$ for $k$ an Integer $>0$. Let

$$p(d)\equiv \max p(d,a)$$

such that $1\leq a<d$ and $(a,d)=1$. Then there exists a $d_0\geq 2$ and an $L>1$ such that $p(d)<d^L$ for all $d>d_0$. $L$ is known as Linnik's Constant.

References

Linnik, U. V. ``On the Least Prime in an Arithmetic Progression. I. The Basic Theorem.'' Mat. Sbornik N. S. 15 (57), 139-178, 1944.

Linnik, U. V. ``On the Least Prime in an Arithmetic Progression. II. The Deuring-Heilbronn Phenomenon'' Mat. Sbornik N. S. 15 (57), 347-368, 1944.