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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]]
An underweight individual may have low immunity, reduced stamina, along with a deficiency of vitamins in his body. So, fat control is rather important. There are numerous ways to determine the perfect weight of the person. Among them are, waist to height ratio, and height to weight ratio.<br><br>Once I went from my cut down plus got rid of all the excess weight I ended up right where the [http://safedietplansforwomen.com/bmi-chart bmi chart] expected me to be at the significant end of the normal range... which makes perfect sense because I've built because much muscle because I may without drugs.<br><br>Now let's have a peep into childhood obesity facts. Children commonly inherit this condition from their parents. The number of overweight children is improving at an alarming rate. As compared to the last three years bmi chart men, this number has doubled. Sedentary activities which are preferred by kids create them dull and lazy. They tend to take foods which lack inside nutritional values. As there is no physical exertion involved, the fats get deposited inside body plus make them lethargic. The general capability of fat youngsters is very low in comparison to others. One of the other childhood weight details is the fact that such children suffer from several additional difficulties like lack of concentration, inability to rest properly, plus humiliation faced from peers and friends.<br><br>One method is to observe their weight plus keep it. However weight is not truly the only factor which decides whether a female is overweight or no. A BMI Calculator helps a female measure her body mass index. This informs her what her perfect fat about her height is. A BMI Calculator female has become an indispensable tool for ladies.<br><br>BMI Calculator is nothing however a Body mass index calculator. It assists ladies track their fat reduction system in a greater way than really viewing the weighing scales. Women inside their battle of the bulge, forget that muscle also has considerable fat and perfect weight is constantly deduced in terms of a woman's height. A taller female may weigh over a smaller female. At the same time, a woman whom apparently looks skinny might weigh more than another who looks stouter than her. This is thus considering bones plus muscles moreover contribute to a body mass and weight. This is another reason why BMI for females is a greater index than fat or vital measurements.<br><br>The U.S. Department of Health plus Human Services points out that Blacks have less 'bad fat' than whites. "Blacks tend to carry about less of a very harmful bmi chart women type of abdominal fat than whites, despite the reality they suffer more from obesity-linked illness, researchers report" say the report. Normally, a person with a 30 BMI is obese along with a individual with a 23 BMI is obese. There are no official BMI numbers for black to use because a guideline. The medical establishment is probably reluctant to raise the numbers because it might provide people a green light to consume more.<br><br>Nonetheless, it utilizes the U.S. Navy Circumference Method which will need the input of information for it to be capable to solve for the body fat percentage. The readings could be put into a body fat chart for the person to be able to monitor the decrease or increase of body fat percentage.<br><br>You are able to lose weight. Consistency is everything. Soon you'll start to see the pounds dropping off. Before we understand it, you'll be a slimmer, healthier adaptation of oneself.
 
:<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 19:16, 10 February 2014

An underweight individual may have low immunity, reduced stamina, along with a deficiency of vitamins in his body. So, fat control is rather important. There are numerous ways to determine the perfect weight of the person. Among them are, waist to height ratio, and height to weight ratio.

Once I went from my cut down plus got rid of all the excess weight I ended up right where the bmi chart expected me to be at the significant end of the normal range... which makes perfect sense because I've built because much muscle because I may without drugs.

Now let's have a peep into childhood obesity facts. Children commonly inherit this condition from their parents. The number of overweight children is improving at an alarming rate. As compared to the last three years bmi chart men, this number has doubled. Sedentary activities which are preferred by kids create them dull and lazy. They tend to take foods which lack inside nutritional values. As there is no physical exertion involved, the fats get deposited inside body plus make them lethargic. The general capability of fat youngsters is very low in comparison to others. One of the other childhood weight details is the fact that such children suffer from several additional difficulties like lack of concentration, inability to rest properly, plus humiliation faced from peers and friends.

One method is to observe their weight plus keep it. However weight is not truly the only factor which decides whether a female is overweight or no. A BMI Calculator helps a female measure her body mass index. This informs her what her perfect fat about her height is. A BMI Calculator female has become an indispensable tool for ladies.

BMI Calculator is nothing however a Body mass index calculator. It assists ladies track their fat reduction system in a greater way than really viewing the weighing scales. Women inside their battle of the bulge, forget that muscle also has considerable fat and perfect weight is constantly deduced in terms of a woman's height. A taller female may weigh over a smaller female. At the same time, a woman whom apparently looks skinny might weigh more than another who looks stouter than her. This is thus considering bones plus muscles moreover contribute to a body mass and weight. This is another reason why BMI for females is a greater index than fat or vital measurements.

The U.S. Department of Health plus Human Services points out that Blacks have less 'bad fat' than whites. "Blacks tend to carry about less of a very harmful bmi chart women type of abdominal fat than whites, despite the reality they suffer more from obesity-linked illness, researchers report" say the report. Normally, a person with a 30 BMI is obese along with a individual with a 23 BMI is obese. There are no official BMI numbers for black to use because a guideline. The medical establishment is probably reluctant to raise the numbers because it might provide people a green light to consume more.

Nonetheless, it utilizes the U.S. Navy Circumference Method which will need the input of information for it to be capable to solve for the body fat percentage. The readings could be put into a body fat chart for the person to be able to monitor the decrease or increase of body fat percentage.

You are able to lose weight. Consistency is everything. Soon you'll start to see the pounds dropping off. Before we understand it, you'll be a slimmer, healthier adaptation of oneself.