Let $K$ be a number field and let $O_K$ its ring of integers. Identify $K$ with its image in $\mathbb{C}^{\text{Hom}_{\mathbb{Q}-\text{alg}}(K,\mathbb{C})}$, which we consider equipped with the $|| \cdot ||_{\infty}$-norm, i.e. the largest absolute value of all embeddings. For a positive real number $X$, denote by $\pi_K(X)$ the number of elements $\alpha \in O_K$ such that $|| \alpha ||_{\infty} \leq X$ and $\alpha$ generates a prime ideal.

Question: What is the asymptotic growth of $\pi_K(X)$ as $X$ goes to infinity? Is there a precise asymptotic formula (with some explicit bound for the error term)?

I wonder if the question (especially the second) can be approached by means of some suitable zeta-function, which instead of being defined over the ideal group, takes into account also the presence of units.

Generalized prime number theorem(Japanese J. Math, 1956). This has the classical de la Vallee Poussin error term, regarding K as fixed, and includes your question as a very particular case. The zeta functions you were looking for are the unramified Grossencharacter Hecke $L$-functions over $K$. $\endgroup$ – Vesselin Dimitrov Feb 4 '18 at 23:08