# Linnik's theorem

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 {displaystyle 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: {displaystyle operatorname {p} (a,d)-functions". Sci. Sinica. 20 (5): 529–562. MR 0476668. ^ Jutila, Matti (1977). "On Linnik's constant". Math. Scand. 41 (1): 45–62. doi:10.7146/math.scand.a-11701. MR 0476671. ^ Graham, Sidney West (1977). Applications of sieve methods (Ph.D.). Ann Arbor, Mich: Univ. Michigan. MR 2627480. ^ Graham, S. W. (1981). "On Linnik's constant". Acta Arith. 39 (2): 163–179. doi:10.4064/aa-39-2-163-179. MR 0639625. ^ Chen, Jingrun (1979). "On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet's -functions. II". Sci. Sinica. 22 (8): 859–889. MR 0549597. ^ Chen, Jingrun; Liu, Jian Min (1989). "On the least prime in an arithmetical progression. III". Science in China Series A: Mathematics. 32 (6): 654–673. MR 1056044. ^ Chen, Jingrun; Liu, Jian Min (1989). "On the least prime in an arithmetical progression. IV". Science in China Series A: Mathematics. 32 (7): 792–807. MR 1058000. ^ Wang, Wei (1991). "On the least prime in an arithmetical progression". Acta Mathematica Sinica. New Series. 7 (3): 279–288. doi:10.1007/BF02583005. MR 1141242. S2CID 121701036. ^ Xylouris, Triantafyllos (2011). "On Linnik's constant". Acta Arith. 150 (1): 65–91. doi:10.4064/aa150-1-4. MR 2825574. ^ Xylouris, Triantafyllos (2011). Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression [The zeros of Dirichlet L-functions and the least prime in an arithmetic progression] (Dissertation for the degree of Doctor of Mathematics and Natural Sciences) (in German). Bonn: Universität Bonn, Mathematisches Institut. MR 3086819. Categories: Theorems in analytic number theoryTheorems about prime numbers

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