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:

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

The theorem is named after Yuri Vladimirovich Linnik, who proved it in 1944.^[1][2] 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) ≤ 2^d − 1, for all d ≥ 3. It is known that p(1,p) ≤ L_p, for all primes p ≥ 5, as L_p is congruent to 1 modulo p for all prime numbers p, where L_p 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.^[3]

On the generalized Riemann hypothesis it can be shown that

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

where ${\displaystyle \varphi }$  is the totient function,^[4] and the stronger bound

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

has been also proved.^[5]

It is also conjectured that:

${\displaystyle \operatorname {p} (a,d)<d^{2}.\;}$  ^[4]

Bounds for L

The constant L is called Linnik's constant^[6] and the following table shows the progress that has been made on determining its size.

L ≤	Year of publication	Author
10000	1957	Pan^[7]
5448	1958	Pan
777	1965	Chen^[8]
630	1971	Jutila
550	1970	Jutila^[9]
168	1977	Chen^[10]
80	1977	Jutila^[11]
36	1977	Graham^[12]
20	1981	Graham^[13] (submitted before Chen's 1979 paper)
17	1979	Chen^[14]
16	1986	Wang
13.5	1989	Chen and Liu^[15]^[16]
8	1990	Wang^[17]
5.5	1992	Heath-Brown^[4]
5.18	2009	Xylouris^[18]
5	2011	Xylouris^[19]

Moreover, in Heath-Brown's result the constant c is effectively computable.

Notes

1. Linnik, Yu. V. (1944). "On the least prime in an arithmetic progression I. The basic theorem". Rec. Math. (Mat. Sbornik). Nouvelle Série. 15 (57): 139–178. MR 0012111.
2. Linnik, Yu. V. (1944). "On the least prime in an arithmetic progression II. The Deuring-Heilbronn phenomenon". Rec. Math. (Mat. Sbornik). Nouvelle Série. 15 (57): 347–368. MR 0012112.
3. Bombieri, Enrico; Friedlander, John B.; Iwaniec, Henryk (1989). "Primes in Arithmetic Progressions to Large Moduli. III". Journal of the American Mathematical Society. 2 (2): 215–224. doi:10.2307/1990976. JSTOR 1990976. MR 0976723.
4. Heath-Brown, Roger (1992). "Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression". Proc. London Math. Soc. 64 (3): 265–338. doi:10.1112/plms/s3-64.2.265. MR 1143227.
5. Lamzouri, Y.; Li, X.; Soundararajan, K. (2015). "Conditional bounds for the least quadratic non-residue and related problems". Math. Comp. 84 (295): 2391–2412. arXiv:1309.3595. doi:10.1090/S0025-5718-2015-02925-1. S2CID 15306240.
6. Guy, Richard K. (2004). Unsolved problems in number theory. Problem Books in Mathematics. Vol. 1 (Third ed.). New York: Springer-Verlag. p. 22. doi:10.1007/978-0-387-26677-0. ISBN 978-0-387-20860-2. MR 2076335.
7. Pan, Cheng Dong (1957). "On the least prime in an arithmetical progression". Sci. Record. New Series. 1: 311–313. MR 0105398.
8. Chen, Jingrun (1965). "On the least prime in an arithmetical progression". Sci. Sinica. 14: 1868–1871.
9. Jutila, Matti (1970). "A new estimate for Linnik's constant". Ann. Acad. Sci. Fenn. Ser. A. 471. MR 0271056.
10. Chen, Jingrun (1977). "On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet's $L$-functions". Sci. Sinica. 20 (5): 529–562. MR 0476668.
11. Jutila, Matti (1977). "On Linnik's constant". Math. Scand. 41 (1): 45–62. doi:10.7146/math.scand.a-11701. MR 0476671.
12. Graham, Sidney West (1977). Applications of sieve methods (Ph.D.). Ann Arbor, Mich: Univ. Michigan. MR 2627480.
13. Graham, S. W. (1981). "On Linnik's constant". Acta Arith. 39 (2): 163–179. doi:10.4064/aa-39-2-163-179. MR 0639625.
14. Chen, Jingrun (1979). "On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet's $L$-functions. II". Sci. Sinica. 22 (8): 859–889. MR 0549597.
15. 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.
16. 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.
17. 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.
18. Xylouris, Triantafyllos (2011). "On Linnik's constant". Acta Arith. 150 (1): 65–91. doi:10.4064/aa150-1-4. MR 2825574.
19. 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.