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

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

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.

Properties

It is known that L ≤ 2 for almost all integers d.[3]

On the generalized Riemann hypothesis it can be shown that

where is the totient function.[4] and the stronger bound

has been also proved.[5]

It is also conjectured that:

[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 publicationAuthor
100001957Pan[7]
54481958Pan
7771965Chen[8]
6301971Jutila
5501970Jutila[9]
1681977Chen[10]
801977Jutila[11]
361977Graham[12]
201981Graham[13] (submitted before Chen's 1979 paper)
171979Chen[14]
161986Wang
13.51989Chen and Liu[15][16]
81990Wang[17]
5.51992Heath-Brown[4]
5.182009Xylouris[18]
52011Xylouris[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) N.S. 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) N.S. 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. 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.