Linnik's theorem

Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem. It asserts that, if we denote p(a,d) the least prime in the arithmetic progression {a + n d}, for integer n>0, where a and d are any given positive coprime integers that 1 ≤ a ≤ d, there exist positive c and L such that:

<math> p(a,d) < c d^{L} \; .</math>

The Theorem is named after Yuri Vladimirovich Linnik[?] (1915-1972) who proved it in 1944.

As of 1992 we know that the Linnik's constant L ≤ 5.5 but we can take L=2 for almost all integers d. It is also conjectured that:

<math> p(a,d) < d \ln^{2} d \; .</math>