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 ideals of a number field K, with norm at most X.

Example

What to expect can be seen already for the Gaussian integers. There for any prime number p of the form 4n + 1, p factors as a product of two Gaussian primes of norm p. Primes of the form 4n + 3 remain prime, giving a Gaussian prime of norm p2. Therefore, we should estimate

2 r ( X ) + r ′ ( X ) {\displaystyle 2r(X)+r^{\prime }({\sqrt {X}})}

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

Y 2 log ⁡ Y . {\displaystyle {\frac {Y}{2\log Y}}.}

Therefore, the 2r(X) term dominates, and is asymptotically

X log ⁡ X . {\displaystyle {\frac {X}{\log X}}.}

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 Landau 1903, for norm at most X the same asymptotic formula

X log ⁡ X {\displaystyle {\frac {X}{\log X}}}

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

L i ( X ) + O K ( X exp ⁡ ( − c K log ⁡ ( X ) ) ) , {\displaystyle \mathrm {Li} (X)+O_{K}(X\exp(-c_{K}{\sqrt {\log(X)}})),\,}

where cK is a constant depending on K.

See also