Euler's four-square identity: Difference between revisions

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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 [[primes in arithmetic progression|prime in the arithmetic progression]]
 
:<math>a + nd,\ </math>
 
where ''n'' runs through the positive [[integer]]s and ''a'' and ''d'' are any given positive [[coprime]] integers with 1 ≤ ''a'' ≤ ''d'' - 1, then:
 
: <math> p(a,d) < c d^{L}. \; </math>
 
The theorem is named after [[Yuri Vladimirovich Linnik]], who proved it in 1944.<ref>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</ref><ref>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</ref> Although Linnik's proof showed ''c'' and ''L'' to be [[effective results in number theory|effectively computable]], he provided no numerical values for them.
 
== Properties ==
 
It is known that ''L'' ≤ 2 for [[almost all]] integers ''d''.<ref>[[Enrico Bombieri|E. Bombieri]], [[John Friedlander|J. B. Friedlander]], [[Henryk Iwaniec|H. Iwaniec]]. "Primes in Arithmetic Progressions to Large Moduli. III", ''Journal of the American Mathematical Society'' '''2'''(2) (1989), pp. 215–224.</ref>
 
On the [[generalized Riemann hypothesis]] it can be shown that
 
: <math> p(a,d) \leq (1+o(1))\varphi(d)^2 \ln^2 d \; ,</math>
 
where <math>\varphi</math> is the [[totient function]].<ref name="heath-brown"/>
 
It is also conjectured that:
 
: <math> p(a,d) < d^2. \; </math> <ref name="heath-brown"/>
 
 
== 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.
 
{| cellpadding="3"
| L ≤ || Year of publication || Author
|-
| align="right" | 10000 || align="center" | 1957 || [[Pan Chengdong|Pan]]<ref>Pan Cheng Dong ''On the least prime in an arithmetical progression.'' Sci. Record (N.S.) 1 (1957) pp. 311-313</ref>
|-
| align="right" | 5448 || align="center" | 1958 || Pan
|-
| align="right" | 777 || align="center" | 1965 || [[Chen Jingrun|Chen]]<ref>Chen Jingrun ''On the least prime in an arithmetical progression.'' Sci. Sinica '''14''' (1965) pp. 1868-1871</ref>
|-
| align="right" | 630 || align="center" | 1971 || [[Matti Jutila|Jutila]]
|-
| align="right" | 550 || align="center" | 1970  || Jutila<ref>Jutila, M. ''A new estimate for Linnik's constant.'' Ann. Acad. Sci. Fenn. Ser. A I No. 471 (1970) 8 pp.</ref>
|-
| align="right" | 168 || align="center" | 1977 || Chen<ref>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</ref>
|-
| align="right" | 80 || align="center" | 1977 || Jutila<ref>Jutila, M. ''On Linnik's constant.'' Math. Scand. '''41''' (1977), no. 1, pp. 45-62</ref>
|-
| align="right" | 36 || align="center" | 1977 || [[Sidney Graham|Graham]]<ref>''Applications of sieve methods'' Ph.D. Thesis, Univ. Michigan, Ann Arbor, Mich., 1977</ref>
|-
| align="right" | 20 || align="center" | 1981 || Graham<ref>Graham, S. W. ''On Linnik's constant.'' Acta Arith. '''39''' (1981), no. 2, pp. 163-179</ref> (submitted before Chen's 1979 paper)
|-
| align="right" | 17 || align="center" | 1979 || Chen<ref>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</ref>
|-
| align="right" | 16 || align="center" | 1986 || Wang
|-
| align="right" | 13.5 || align="center" | 1989 || Chen and [[Liu Jian Min|Liu]]<ref>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</ref><ref>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</ref>
|-
| align="right" | 8 || align="center" | 1990 || Wang<ref>Wang ''On the least prime in an arithmetical progression. '' Acta Mathematica Sinica, New Series 1991 Vol. 7 No. 3 pp. 279-288</ref>
|-
| align="right" |  5.5 || align="center" | 1992 || [[Roger Heath-Brown|Heath-Brown]]<ref name="heath-brown">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</ref>
|-
| align="right" | 5.2 || align="center" | 2009 || Xylouris<ref>Triantafyllos Xylouris, On Linnik's constant (2009). {{arXiv|0906.2749}}</ref>
|-
| align="right" | 5 || align="center" | 2011 || Xylouris<ref>Triantafyllos Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression (2011). Dr. rer. nat. dissertation.</ref>
|}
 
Moreover, in Heath-Brown's result the constant ''c'' is effectively computable.
 
==Notes==
{{reflist|colwidth=30em}}
 
[[Category:Theorems in analytic number theory]]
[[Category:Theorems about prime numbers]]

Revision as of 00:09, 14 August 2013

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 ≤ a ≤ d - 1, then:

p(a,d)<cdL.

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

p(a,d)≤(1+o(1))φ(d)2ln2d,

where φ is the totient function.[4]

It is also conjectured that:

p(a,d)<d2. [4]


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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  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.
  4. ↑ 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
  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). Template:ArXiv
  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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