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


 * $$a + nd,\ $$

where n runs through the positive integers and a and d are any given positive coprime integers with 1 ≤ a ≤ d − 1, then:


 * $$\operatorname{p}(a,d) < c d^{L}. \; $$

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 follows from Zsigmondy's theorem that p(1,d) ≤ 2d − 1, for all d ≥ 3. It is known that p(1,p) ≤ Lp, for all primes p ≥ 5, as Lp is congruent to 1 modulo p for all prime numbers p, where Lp denotes the p-th Lucas number. Just like Mersenne numbers, Lucas numbers with prime indices have divisors of the form 2kp+1.

Properties
It is known that L ≤ 2 for almost all integers d.

On the generalized Riemann hypothesis it can be shown that


 * $$\operatorname{p}(a,d) \leq (1+o(1))\varphi(d)^2 (\log d)^2 \; ,$$

where $$\varphi$$ is the totient function, and the stronger bound


 * $$\operatorname{p}(a,d) \leq \varphi(d)^2 (\log d)^2 \; ,$$

has been also proved.

It is also conjectured that:


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

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.