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>