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(a,d) 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.
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.
It is also conjectured that:
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.
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