 Linnik's theorem - Wikiwand

# 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

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

$p(a,d) The theorem is named after Yuri Vladimirovich Linnik, who proved it in 1944. 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.

On the generalized Riemann hypothesis it can be shown that

$p(a,d)\leq (1+o(1))\varphi (d)^{2}\ln ^{2}d\;,$ where $\varphi$ is the totient function. and the stronger bound

$p(a,d)\leq \varphi (d)^{2}\ln ^{2}d\;,$ has been also proved.

It is also conjectured that:

$p(a,d) ## Bounds for L

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

 L ≤ Year of publication Author 10000 1957 Pan 5448 1958 Pan 777 1965 Chen 630 1971 Jutila 550 1970 Jutila 168 1977 Chen 80 1977 Jutila 36 1977 Graham 20 1981 Graham (submitted before Chen's 1979 paper) 17 1979 Chen 16 1986 Wang 13.5 1989 Chen and Liu 8 1990 Wang 5.5 1992 Heath-Brown 5.18 2009 Xylouris 5 2011 Xylouris

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

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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.