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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''.
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| 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
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| :<math>2r(X)+r^\prime(\sqrt{X})</math>
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| 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
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| :<math>\frac{Y}{2\log Y}.</math>
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| Therefore the 2''r''(''X'') term predominates, and is asymptotically
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| :<math>\frac{X}{\log X}.</math> | |
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| 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
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| :<math>\frac{X}{\log X}</math> | |
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| 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.
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| 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
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| :<math> \mathrm{Li}(X) + O_K(X \exp(-c_K \sqrt{\log(X)}) , \,</math>
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| where ''c''<sub>''K''</sub> is a constant depending on ''K''.
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| ==See also==
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| * [[Abstract analytic number theory]]
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| ==References==
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| * {{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 }}
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| * {{Cite journal
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| | doi=10.1007/BF01444310
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| | last=Landau
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| | first=Edmund
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| | author-link=Edmund Landau
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| | title=Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes
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| | year=1903
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| | journal=Mathematische Annalen
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| | volume=56
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| | issue=4
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| | pages=645–670
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| | ref=harv
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| }}
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| * {{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}}
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| [[Category:Theorems in analytic number theory]]
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| [[Category:Theorems in algebraic number theory]]
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