Euler's four-square identity
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]
It is also conjectured that:
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[5] |
| 5448 | 1958 | Pan |
| 777 | 1965 | Chen[6] |
| 630 | 1971 | Jutila |
| 550 | 1970 | Jutila[7] |
| 168 | 1977 | Chen[8] |
| 80 | 1977 | Jutila[9] |
| 36 | 1977 | Graham[10] |
| 20 | 1981 | Graham[11] (submitted before Chen's 1979 paper) |
| 17 | 1979 | Chen[12] |
| 16 | 1986 | Wang |
| 13.5 | 1989 | Chen and Liu[13][14] |
| 8 | 1990 | Wang[15] |
| 5.5 | 1992 | Heath-Brown[4] |
| 5.2 | 2009 | Xylouris[16] |
| 5 | 2011 | Xylouris[17] |
Moreover, in Heath-Brown's result the constant c is effectively computable.
Notes
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- ↑ Linnik, Yu. V. On the least prime in an arithmetic progression I. The basic theorem Rec. Math. (Mat. Sbornik) N.S. 15 (57) (1944), pages 139-178
- ↑ Linnik, Yu. V. On the least prime in an arithmetic progression II. The Deuring-Heilbronn phenomenon Rec. Math. (Mat. Sbornik) N.S. 15 (57) (1944), pages 347-368
- ↑ E. Bombieri, J. B. Friedlander, H. Iwaniec. "Primes in Arithmetic Progressions to Large Moduli. III", Journal of the American Mathematical Society 2(2) (1989), pp. 215–224.
- ↑ 4.0 4.1 4.2 Heath-Brown, D. R. Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64(3) (1992), pp. 265-338
- ↑ Pan Cheng Dong On the least prime in an arithmetical progression. Sci. Record (N.S.) 1 (1957) pp. 311-313
- ↑ Chen Jingrun On the least prime in an arithmetical progression. Sci. Sinica 14 (1965) pp. 1868-1871
- ↑ Jutila, M. A new estimate for Linnik's constant. Ann. Acad. Sci. Fenn. Ser. A I No. 471 (1970) 8 pp.
- ↑ Chen Jingrun On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet's $L$-functions. Sci. Sinica 20 (1977), no. 5, pp. 529-562
- ↑ Jutila, M. On Linnik's constant. Math. Scand. 41 (1977), no. 1, pp. 45-62
- ↑ Applications of sieve methods Ph.D. Thesis, Univ. Michigan, Ann Arbor, Mich., 1977
- ↑ Graham, S. W. On Linnik's constant. Acta Arith. 39 (1981), no. 2, pp. 163-179
- ↑ Chen Jingrun On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet's $L$-functions. II. Sci. Sinica 22 (1979), no. 8, pp. 859-889
- ↑ Chen Jingrun and Liu Jian Min On the least prime in an arithmetical progression. III. Sci. China Ser. A 32 (1989), no. 6, pp. 654-673
- ↑ Chen Jingrun and Liu Jian Min On the least prime in an arithmetical progression. IV. Sci. China Ser. A 32 (1989), no. 7, pp. 792-807
- ↑ Wang On the least prime in an arithmetical progression. Acta Mathematica Sinica, New Series 1991 Vol. 7 No. 3 pp. 279-288
- ↑ Triantafyllos Xylouris, On Linnik's constant (2009). Template:ArXiv
- ↑ Triantafyllos Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression (2011). Dr. rer. nat. dissertation.