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.
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
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 | |
| 2logY |
.
Therefore, the 2r(X) term dominates, and is asymptotically
| X | |
| logX |
.
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, for norm at most X the same asymptotic formula
| X | |
| logX |
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
Li(X)+OK(X\exp(-cK\sqrt{log(X)})),
where cK is a constant depending on K.
. Hugh L. Montgomery . Hugh Montgomery (mathematician) . Robert C. Vaughan . Robert Charles Vaughan (mathematician) . Multiplicative number theory I. Classical theory . Cambridge tracts in advanced mathematics . 97 . 2007 . 978-0-521-84903-6 . 266–268.