Generalization of the prime number theorem to other fields

Question

Is there a similar statement to the prime number theorem in other rings like $\mathbb{Z}[i]$ or $\mathbb{Z}[\omega]$.

Yes, see http://en.wikipedia.org/wiki/Landau_prime_ideal_theorem — tc1729, Feb 08 '14 at 00:28
Technicality: ${\bf Z}[i]$ is not a field, a fortiori, not a number field. It is the ring of integers in ${\bf Q}[i]$, which is a number field. Similarly for ${\bf Z}[\omega]$. — Gerry Myerson, Feb 08 '14 at 00:49

score 4 · Accepted Answer · answered Feb 08 '14 at 01:53

Yes, if $\mathcal O_K$ is the ring of integers in a number field $K$, and $\pi_K(x)$ denotes the number of non-zero primes ideals in $\mathcal O_K$ of norm $\leq x$, then $\pi_K(x) \sim x/\log x.$

The link in Siddharth Prasad's comment has more details. The proof uses the same $\zeta$-function techniques as the proof of the usual prime number theorem.

Generalization of the prime number theorem to other fields

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