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en>Loraof: /* Triangle method */ no italics for words in equations

2014-12-31T18:42:51Z

Triangle method: no italics for words in equations

← Older revision		Revision as of 20:42, 31 December 2014
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	~~In [[number theory]], the '''Elliott–Halberstam conjecture'''~~ is ~~a [[conjecture]] about~~ the ~~distribution~~ of ~~[[prime number]]~~s in ~~[[arithmetic progression]]s~~. ~~It has~~ many ~~applications in [[sieve theory]]. It is named for [[Peter D. T. A. Elliott]] and [[Heini Halberstam]].~~		She is recognized by the title of Myrtle Shryock. For a whilst she's been in South Dakota. For many years he's been operating as a meter reader and it's something he really appreciate. To collect coins is a thing that I'm completely addicted to.<br><br>Also visit my web-site std testing at home ([http://dore.gia.ncnu.edu.tw/88ipart/node/1857887 visit the following website page])

	~~To state the conjecture requires some notation. Let <math>\pi(x)</math> denote the number of primes less than or equal to~~ '~~'x''. If ''q'' is~~ a ~~[[negative and positive numbers\|positive]] [[integer]]~~ and '~~'a''~~ is ~~[[coprime]] to ''q'', we let <math>\pi(x;q,~~a~~)</math>, denote the number of primes less than or equal to~~ '~~'x'' which are equal~~ to ~~''a'' modulo ''q''~~. ~~[[Dirichlet's theorem on primes in arithmetic progressions]] then tells us~~
	~~that~~

	~~:<math> \pi(x;q,a) \approx \frac{\pi(x)}{\varphi(q)}~~<~~/math~~>

	~~where ''a'' is coprime to ''q'' and <math>\varphi</math> is [[Euler's totient function]]. If we then define the error function~~

	:<~~math~~> ~~E(x;q) = \max_{(a,q) = 1} \left\|\pi(x;q,a)~~ - ~~\frac{\pi~~(~~x)}{\varphi(q)}\right\|</math>~~

	~~where the max is taken over all ''a'' coprime to ''q'', then the Elliott–Halberstam conjecture is the assertion that~~
	~~for every ''θ'' < 1 and ''A'' > 0 there exists a constant ''C'' > 0 such that~~

	:~~<math> \sum_{1 \leq q \leq x^\theta} E(x;q) \leq \frac{C x}{\log^A x}<~~/~~math>~~

	~~for all ''x'' > 2.~~

	~~This conjecture was proven for all ''θ'' < 1~~/~~2 by [[Enrico Bombieri]] and [[A~~. I. Vinogradov]] (the [[Bombieri–Vinogradov theorem]], sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the [[generalized Riemann hypothesis]]. ~~It is known that the conjecture fails at the endpoint ''θ'' = 1~~.

	~~The Elliott–Halberstam conjecture has several consequences~~. ~~One striking one is the result announced by [[Dan Goldston]], [[János Pintz]], and [[Cem Yıldırım]],<ref>{{arxiv\|math.NT~~/~~0508185}}; see also {{arxiv\|math.NT~~/~~0505300}}, {{arxiv\|math.NT~~/0506067}}.</ref> which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013, James Maynard showed that subject to the Elliott-Halberstam conjecture, one can show the ~~existence of infinitely many pairs of consecutive primes that differ by at most 12.~~

	~~==See also==~~
	*[[Barban–Davenport–Halberstam theorem]]
	*[[Barban–Montgomery theorem]]

	~~==Notes==~~
	~~{{Reflist}}~~

	~~== References ==~~
	*{{cite journal \|first=E. \|last=Bombieri \|title=On the large sieve \|journal=Mathematika \|volume=12 \|issue= \|year=1965 \|pages=201–225 \|doi= }}
	*{{cite journal \|first=P. D. T. A. \|last=Elliott \|first2=H. \|last2=Halberstam \|title=A conjecture in prime number theory \|journal=Symp. Math. \|volume=4 \|year=1968 \|issue= \|pages=59–72 }}
	*{{cite journal \|first=A. I. \|last=Vinogradov \|title=The density hypothesis for Dirichlet L-series \|language=Russian \|journal=Izv. Akad. Nauk SSSR Ser. Mat. \|volume=29 \|issue=4 \|year=1965 \|pages=903–934 \|mr=197414 }}
	*{{cite journal \|first=K. \|last=Soundararajan \|authorlink=Kannan Soundararajan \|title=Small gaps between prime numbers: The work of Goldston–Pintz–Yıldırım \|journal=Bull. AMS \|volume=44 \|year=2007 \|issue=1 \|pages=1–18 \|doi=10.1090/S0273-0979-06-01142-6 }}

	~~{{DEFAULTSORT:Elliott-Halberstam conjecture}}~~
	~~[[Category:Analytic number theory]]~~
	~~[[Category:Conjectures about prime numbers]~~]

112.207.23.172 at 13:09, 18 January 2014

2014-01-18T13:09:59Z

← Older revision		Revision as of 15:09, 18 January 2014
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	I'~~m Michal~~ (28) ~~from Corrylach~~, ~~Great Britain~~. <br>I'~~m learning Dutch literature at~~ a ~~local university~~ and I'~~m just about to graduate~~.<br>~~I have~~ a ~~part time job in~~ a ~~backery.~~<br><br>~~my page~~ :~~: FIFA 15 coin hack~~ ([~~http:~~//~~Dev3~~.~~ittlakom~~.hu/~~tartalom~~/~~how~~-~~get~~-~~free~~-~~fifa~~-15-~~coins-3 click the following post~~])		In [[number theory]], the '''Elliott–Halberstam conjecture''' is a [[conjecture]] about the distribution of [[prime number]]s in [[arithmetic progression]]s. It has many applications in [[sieve theory]]. It is named for [[Peter D. T. A. Elliott]] and [[Heini Halberstam]].

			To state the conjecture requires some notation. Let <math>\pi(x)</math> denote the number of primes less than or equal to ''x''. If ''q'' is a [[negative and positive numbers\|positive]] [[integer]] and ''a'' is [[coprime]] to ''q'', we let <math>\pi(x;q,a)</math>, denote the number of primes less than or equal to ''x'' which are equal to ''a'' modulo ''q''. [[Dirichlet's theorem on primes in arithmetic progressions]] then tells us
			that

			:<math> \pi(x;q,a) \approx \frac{\pi(x)}{\varphi(q)}</math>

			where ''a'' is coprime to ''q'' and <math>\varphi</math> is [[Euler's totient function]]. If we then define the error function

			:<math> E(x;q) = \max_{(a,q) = 1} \left\|\pi(x;q,a) - \frac{\pi(x)}{\varphi(q)}\right\|</math>

			where the max is taken over all ''a'' coprime to ''q'', then the Elliott–Halberstam conjecture is the assertion that
			for every ''θ'' < 1 and ''A'' > 0 there exists a constant ''C'' > 0 such that

			:<math> \sum_{1 \leq q \leq x^\theta} E(x;q) \leq \frac{C x}{\log^A x}</math>

			for all ''x'' > 2.

			This conjecture was proven for all ''θ'' < 1/2 by [[Enrico Bombieri]] and [[A. I. Vinogradov]] (the [[Bombieri–Vinogradov theorem]], sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the [[generalized Riemann hypothesis]]. It is known that the conjecture fails at the endpoint ''θ'' = 1.

			The Elliott–Halberstam conjecture has several consequences. One striking one is the result announced by [[Dan Goldston]], [[János Pintz]], and [[Cem Yıldırım]],<ref>{{arxiv\|math.NT/0508185}}; see also {{arxiv\|math.NT/0505300}}, {{arxiv\|math.NT/0506067}}.</ref> which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013, James Maynard showed that subject to the Elliott-Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 12.

			==See also==
			*[[Barban–Davenport–Halberstam theorem]]
			*[[Barban–Montgomery theorem]]

			==Notes==
			{{Reflist}}

			== References ==
			*{{cite journal \|first=E. \|last=Bombieri \|title=On the large sieve \|journal=Mathematika \|volume=12 \|issue= \|year=1965 \|pages=201–225 \|doi= }}
			*{{cite journal \|first=P. D. T. A. \|last=Elliott \|first2=H. \|last2=Halberstam \|title=A conjecture in prime number theory \|journal=Symp. Math. \|volume=4 \|year=1968 \|issue= \|pages=59–72 }}
			*{{cite journal \|first=A. I. \|last=Vinogradov \|title=The density hypothesis for Dirichlet L-series \|language=Russian \|journal=Izv. Akad. Nauk SSSR Ser. Mat. \|volume=29 \|issue=4 \|year=1965 \|pages=903–934 \|mr=197414 }}
			*{{cite journal \|first=K. \|last=Soundararajan \|authorlink=Kannan Soundararajan \|title=Small gaps between prime numbers: The work of Goldston–Pintz–Yıldırım \|journal=Bull. AMS \|volume=44 \|year=2007 \|issue=1 \|pages=1–18 \|doi=10.1090/S0273-0979-06-01142-6 }}

			{{DEFAULTSORT:Elliott-Halberstam conjecture}}
			[[Category:Analytic number theory]]
			[[Category:Conjectures about prime numbers]]

42.2.130.56: /* External links */

2012-08-11T14:12:30Z

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I'm Michal (28) from Corrylach, Great Britain. <br>I'm learning Dutch literature at a local university and I'm just about to graduate.<br>I have a part time job in a backery.<br><br>my page :: FIFA 15 coin hack ([http://Dev3.ittlakom.hu/tartalom/how-get-free-fifa-15-coins-3 click the following post])