Linniks sats

Inom talteori är Linniks sats ett resultat om primtal i aritmetiska följder. Satsen säger att det finns positiva konstanter c och L så att om vi betecknar med p(a,d) det minsta primtalet i den aritmetiska följden

a+nd\

där n går över alla positiva heltal och a och d är godtyckliga relativt prima positiva heltal med 1 ≤ a ≤ d - 1, är:

p(a,d)<cd^{L}.\;

Satsen är uppkallad efter Jurij Linnik, som bevisade den 1944.^[1]^[2] Även om Linniks bevis visade att c och L är effektivt beräknelig, gav han inga numeriska värden åt dem.

Egenskaper

Det är känt att L ≤ 2 för nästan alla heltal d.^[3]

Under antagande av generaliserade Riemannhypothsen kan man bevisa att

p(a,d)\leq (1+o(1))\varphi (d)^{2}\ln ^{2}d\;

där $\varphi$ är Eulers fi-funktion.^[4]

Man har även förmodat att

p(a,d)<d^{2}.\;

^[4]

Gränser för L

Konstanten L kallas för Linniks konstant. Följande tabell visar framstegen som har gjorts i problemet att hitta övre gränser för den.

L ≤	Publikationsår	Författare
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] (överlämnat före Chens publikation från 1979)
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]

I Heath-Browns resultat är konstanten c effektivt räknebar.

Referenser

Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia, Linnik's theorem, 10 december 2013.

Noter

^ 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.
^ [a b c] 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). arXiv:0906.2749
^ Triantafyllos Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression (2011). Dr. rer. nat. dissertation.

[1] 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

[2] 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

[3] 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.

[heath-brown-4] [a b c] 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

[5] Pan Cheng Dong On the least prime in an arithmetical progression. Sci. Record (N.S.) 1 (1957) pp. 311-313

[6] Chen Jingrun On the least prime in an arithmetical progression. Sci. Sinica 14 (1965) pp. 1868-1871

[7] Jutila, M. A new estimate for Linnik's constant. Ann. Acad. Sci. Fenn. Ser. A I No. 471 (1970) 8 pp.

[8] 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

[9] Jutila, M. On Linnik's constant. Math. Scand. 41 (1977), no. 1, pp. 45-62

[10] Applications of sieve methods Ph.D. Thesis, Univ. Michigan, Ann Arbor, Mich., 1977

[11] Graham, S. W. On Linnik's constant. Acta Arith. 39 (1981), no. 2, pp. 163-179

[12] 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

[13] 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

[14] 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

[15] Wang On the least prime in an arithmetical progression. Acta Mathematica Sinica, New Series 1991 Vol. 7 No. 3 pp. 279-288

[16] Triantafyllos Xylouris, On Linnik's constant (2009). arXiv:0906.2749

[17] Triantafyllos Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression (2011). Dr. rer. nat. dissertation.

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