en>David Eppstein: /* See also */ spaces in formulas

2014-11-13T07:20:05Z

80.71.135.103: See also *Pépin's test (the special case ''k''=1, where one chooses ''a''=3)

2014-02-15T22:39:46Z

See also *Pépin's test (the special case ''k''=1, where one chooses ''a''=3)

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	~~In [[algebraic number theory]]~~, ~~the~~ '~~''prime ideal theorem''' is the~~ [~~[number field]] generalization of the [[prime number theorem]]~~. ~~It provides an asymptotic formula for counting the number of [[prime ideal]]s of a number field ''K'', with [[field norm\|norm]] at most ''X''~~.		Nice to satisfy you, my title is Figures Held although I don't really like being known as like [https://Www.epapyrus.com/xe/Purchase/5258960 std home test] that. [http://Ehealthforum.com/health/sore-throat-after-oral-sex-t138861.html Managing people] has been his std home test working day occupation for a while. Years in the [http://chatbook.biz/blogs/post/922 std home test] past we moved to North Dakota. What I love doing is doing ceramics but I haven't made a over the [http://Menshealth.about.com/od/diseases/a/syphilis.htm counter std] test dime with it.<br><br>my web blog: [http://203.250.78.160/zbxe/?document_srl=810524&mid=gallery&sort_index=regdate&order_type=desc http://203.250.78.160/zbxe/?document_srl=810524&mid=gallery&sort_index=regdate&order_type=desc]

	~~What to expect can be seen already for the [[Gaussian integer~~]]s. ~~There for any prime number ''p'' of the form 4''n'' + 1, ''p'' factors as a product of two~~ [~~[Gaussian prime]]s of norm ''p''~~. ~~Primes of the form 4''n'' + 3 remain prime, giving a Gaussian prime of norm ''p''<sup>2<~~/~~sup>. Therefore we should estimate~~

	~~:<math>2r(X)+r^\prime(\sqrt{X})<~~/~~math>~~

	~~where ''r'' counts primes in the arithmetic progression 4''n'' + 1, and ''r''′ in the arithmetic progression 4''n'' + 3~~. ~~By the quantitative form of [[Dirichlet's theorem on primes~~]~~], each of~~ ~~''r''(''Y'') and ''r''′(''Y'') is asymptotically~~

	~~:<math>\frac{Y}{2\log Y}~~.~~</math>~~

	~~Therefore~~ the ~~2''r''(''X'') term predominates, and is asymptotically~~

	:~~<math>\frac{X}{\log X}~~.</~~math>~~

	~~This general pattern holds for number fields in general, so that the prime ideal theorem~~ is ~~dominated by~~ the ~~ideals of norm a prime number. As~~ [~~[Edmund Landau]] proved in {{harvnb\|Landau\|1903}}, for norm at most ''X'' the same asymptotic formula~~

	:~~<math>\frac{X}{\log X}<~~/~~math>~~

	~~always holds~~. ~~Heuristically this is because the [[logarithmic derivative]] of the [[Dedekind zeta-function]] of ''K'' always has~~ a ~~simple pole with residue −1 at ''s'' = 1~~.

	As with ~~the Prime Number Theorem, a more precise estimate may be given in terms of the [[logarithmic integral function]]~~. ~~The number of prime ideals of norm ≤ ''X'' is~~

	:<~~math~~> ~~\mathrm{Li}(X) + O_K(X \exp(-c_K \sqrt{\log(X)}) , \,~~</~~math>~~

	~~where ''c''<sub>''K''<~~/~~sub> is a constant depending on ''K''~~.

	~~==See also==~~
	* [[Abstract analytic number theory]]

	~~==References==~~
	* {{cite book \| author=Alina Carmen Cojocaru \| coauthors=[[M. Ram Murty]] \| title=An introduction to sieve methods and their applications \| series=London Mathematical Society Student Texts \| volume=66 \| publisher=[[Cambridge University Press]] \| isbn=0-521-61275-6 \| pages=35–38 }}
	* {{Cite journal
	~~\| doi=10~~.~~1007~~/~~BF01444310~~
	~~\| last~~=~~Landau~~
	~~\| first~~=~~Edmund~~
	~~\| author-link~~=~~Edmund Landau~~
	~~\| title~~=~~Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes~~
	~~\| year=1903~~
	~~\| journal=Mathematische Annalen~~
	~~\| volume=56~~
	~~\| issue=4~~
	~~\| pages=645–670~~
	~~\| ref=harv~~
	}}
	* {{cite book \| author=Hugh L. ~~Montgomery \| authorlink=Hugh Montgomery (mathematician) \| coauthors=[[Robert Charles Vaughan (mathematician)\|Robert C~~. ~~Vaughan]] \| title=Multiplicative number theory I~~. ~~Classical theory \| series~~=~~Cambridge tracts in advanced mathematics \| volume~~=~~97 \| year~~=~~2007 \| isbn=0-521-84903-9 \| pages~~=~~266–268}}~~

	~~[[Category:Theorems in analytic number theory]]~~
	~~[[Category:Theorems in algebraic number theory]~~]

en>Bubba73: /* Numerical examples */

2013-08-09T03:45:37Z

Numerical examples

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89.12.33.98 at 15:48, 24 June 2012

2012-06-24T15:48:16Z

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Proth's theorem - Revision history

en>David Eppstein: /* See also */ spaces in formulas

80.71.135.103: See also *Pépin's test (the special case ''k''=1, where one chooses ''a''=3)

en>Bubba73: /* Numerical examples */

89.12.33.98 at 15:48, 24 June 2012