Landau prime ideal theorem

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

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

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

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

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

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

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

${\displaystyle {\frac {X}{\log X}}}$

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

${\displaystyle \mathrm {Li} (X)+O_{K}(X\exp(-c_{K}{\sqrt {\log(X)}})),\,}$

where c_K is a constant depending on K.

• Abstract analytic number theory

• Alina Carmen Cojocaru; M. Ram Murty (8 December 2005). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. pp. 35–38. ISBN 0-521-61275-6.
• Landau, Edmund (1903). "Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes". Mathematische Annalen. 56 (4): 645–670. doi:10.1007/BF01444310. S2CID 119669682.
• Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. pp. 266–268. ISBN 978-0-521-84903-6.