It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
By the quantitative form of Dirichlet's theorem on primes, each of r(Y) and r′(Y) is asymptotically Therefore, the 2r(X) term dominates, and is asymptotically 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.
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 where cK is a constant depending on K.