Euler's four-square identity: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>Michael Hardy
punctuation correction required by WP:MOS
 
en>Titus III
Re-directed "bilinear" wiki-link to the more appropriate "bilinear map".
Line 1: Line 1:
We understand or your doctor has told you which you have hemorrhoids: today what is the best hemorrhoid treatment. What is how to receive rid of hemorrhoids?<br><br>The second [http://hemorrhoidtreatmentfix.com/thrombosed-hemorrhoid-treatment thrombosed external hemorrhoid treatment] is in the shape of suppository. We have to insert the drug inside the rectum in order to help healing the symptom. This system is considered to be advantageous since countless sufferers have chosen it plus discover it functions effectively with them.<br><br>It is a truth that hemorrhoid is considered to be a shape of vein swelling, which arises about the lower rectal areas. Besides, it happens to be said that forty % of the adults are having hemorrhoids also.<br><br>Step 6 - Try A Suppository. The cheapest thing in this category is petroleum jelly. Many of the suppositories to be had over the counter are made from the same type of ingredients. This means you can benefit from an inexpensive generic brand plus get the same healing relief.<br><br>Another tip which would supply you with some immediate relief to the hemorrhoid issue is chatting a nice warm bathtub. The bath will sooth the pain we feel, plus when you add a little salt into the bath, about a teaspoon or so, plus massage the hemorrhoid this might even further aid with pain relief.<br><br>Since the largest cause of hemorrhoids is strained bowel movements plus hard stools (chronic constipation), countless folks may discover extended term relief from hemorrhoids by acquiring a solutions which allows them to "go" more frequently and easily. Chronic constipation is caused inside desirable piece considering as a society you are a fast food nation. We eat lots of processed food, and not nearly enough all-natural or fresh foods. I challenge we to take a look at the labels on your food for "nutritional fiber content" should you suffer from irregularity. My guess is the fact that there are on many foods you eat the nutritional fiber content to be low.<br><br>I learn this may sound like a lot of water, yet when you are experiencing a hemorrhoid you have to try to drink at least 1 full gallon of water a day. If you can't do this, commence off with half a gallon and move up from there. This will help prevent constipation.
'''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 01:09, 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

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:

[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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  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.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Euler%27s_four-square_identity&oldid=3109"