Linnik's theorem
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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:
- <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>fr:Théorème de Linnik