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Linnik's theorem

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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 ≤ ad - 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 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) 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. ^ a b c 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.
  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.
  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.
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Linnik's theorem
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