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.