Landau prime ideal theorem

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

Example[edit]

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

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

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

\frac{X}{\log X}.

General number fields[edit]

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

\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

 \mathrm{Li}(X) + O_K(X \exp(-c_K \sqrt{\log(X)}) , \,

where cK is a constant depending on K.

See also[edit]

References[edit]