Landau prime ideal theorem
- Landau prime ideal theorem
In mathematics, the prime ideal theorem of algebraic number theory 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 field. There for any prime number "p" of the form 4"n" + 1, "p" factors as a product of two Gaussian primes of norm "p". Primes of the form 4"n" + 3 remain prime, giving a Gaussian prime of norm "p"2. Therefore we should estimate
:
where "r" counts primes in the arithmetic progression 4"n" + 1, and "r"′ in the arithmetic progression 4"n" + 3. By the quantitative form of Dirichlet's theorem on primes, each of "r"("Y") and "r"′("Y") is asymptotically
:
Therefore the 2"r"("X") term predominates, 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. As Edmund Landau proved, for norm at most "X" the same asymptotic formula
:
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
:
where "c""K" is a constant depending on "K".
ee also
* Abstract analytic number theory
* Boolean prime ideal theorem
References
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