Linnik's theorem


Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive c and L such that, if we denote p the least prime in the arithmetic progression
where n runs through the positive integers and a and d are any given positive coprime integers with 1 ≤ a ≤ d - 1, then:
The theorem is named after Yuri Vladimirovich Linnik, who proved it in 1944. Although Linnik's proof showed c and L to be effectively computable, he provided no numerical values for them.

Properties

It is known that L ≤ 2 for almost all integers d.
On the generalized Riemann hypothesis it can be shown that
where is the totient function.
and the stronger bound
has been also proved.
It is also conjectured that:

Bounds for ''L''

The constant L is called Linnik's constant and the following table shows the progress that has been made on determining its size.
Moreover, in Heath-Brown's result the constant c is effectively computable.