Linnik's theorem

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Short description: Mathematical 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

[math]\displaystyle{ a + nd,\ }[/math]

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

[math]\displaystyle{ \operatorname{p}(a,d) \lt c d^{L}. \; }[/math]

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) ≤ 2d − 1, for all d ≥ 3. It is known that p(1,p) ≤ Lp, for all primes p ≥ 5, as Lp is congruent to 1 modulo p for all prime numbers p, where Lp 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

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

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

[math]\displaystyle{ \operatorname{p}(a,d) \leq \varphi(d)^2 (\log d)^2 \; , }[/math]

has been also proved.[5]

It is also conjectured that:

[math]\displaystyle{ \operatorname{p}(a,d) \lt d^2. \; }[/math] [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. 
  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. 
  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. 
  4. 4.0 4.1 4.2 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. https://ora.ox.ac.uk/objects/uuid:b63b8b4f-ad21-4de4-a86b-c82fdfc87997. 
  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. 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. 
  7. Pan, Cheng Dong (1957). "On the least prime in an arithmetical progression". Sci. Record. New Series 1: 311–313. 
  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. 
  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. 
  11. Jutila, Matti (1977). "On Linnik's constant". Math. Scand. 41 (1): 45–62. doi:10.7146/math.scand.a-11701. 
  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. 
  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. 
  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. 
  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. 
  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. 
  18. Xylouris, Triantafyllos (2011). "On Linnik's constant". Acta Arith. 150 (1): 65–91. doi:10.4064/aa150-1-4. 
  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 Deutsch). Bonn: Universität Bonn, Mathematisches Institut. MR 3086819.