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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''.

==Example==
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''&prime; in the arithmetic progression 4''n'' + 3. By the quantitative form of [[Dirichlet's theorem on primes]], each of  ''r''(''Y'') and ''r''&prime;(''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>

==General number fields==
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 &minus;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 &le; ''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|author1-link= Alina Carmen Cojocaru |author2=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 |author2-link= M. Ram Murty }}
* {{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
| url=https://zenodo.org/record/1735140
}}
* {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |authorlink2=Robert Charles Vaughan (mathematician)  | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=266–268}}

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