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

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

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

:$\backslash frac\{Y\}\{2\backslash log\; Y\}.$

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

:$\backslash frac\{X\}\{\backslash log\; X\}.$

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 {{harvnb|Landau|1903}}, for norm at most X the same asymptotic formula

:$\backslash frac\{X\}\{\backslash log\; X\}$

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

:$\backslash mathrm\{Li\}(X)\; +\; O\_K(X\; \backslash exp(-c\_K\; \backslash sqrt\{\backslash log(X)\})),\; \backslash ,$

where c_K is a constant depending on K.

• Abstract analytic number theory

• {{cite book | author=Alina Carmen Cojocaru|author1-link= Alina Carmen Cojocaru |author2=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts |date= 8 December 2005 | volume=66 | publisher=Cambridge University Press | isbn=0-521-61275-6 | pages=35–38 |author2-link= M. Ram Murty }}
• {{Cite journal

| doi=10.1007/BF01444310

| last=Landau

| first=Edmund

| author-link=Edmund Landau

| title=Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes

| year=1903

| journal=Mathematische Annalen

| volume=56

| issue=4

| pages=645–670

| s2cid=119669682

| url=https://zenodo.org/record/1735140

}}

• {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |authorlink2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=266–268}}

Category:Theorems in analytic number theory

Category:Theorems in algebraic number theory