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 + nd,

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:

$p(a,d)<cd^{L}\;.$

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:

$p(a,d)<d\ln ^{2}d\;.$