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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.

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    $\begingroup$ Closely related: Landau prime ideal theorem. $\endgroup$ – Wojowu Feb 4 '18 at 11:15
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    $\begingroup$ You may consider more generally the estimation of prime elements in regions other than the sup balls, looking into the spatial distribution of the prime elements in $K \otimes \mathbb{R}$. The general result for that is in section 4 of Mitsui's paper 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
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    $\begingroup$ @VesselinDimitrov: Many thanks for the reference! That paper is precisely what I need (the problem stated here is a toy version of the type of results I am looking for which are slightly messier to state, but by looking at your reference I have little doubt that the results or at least the methods of Mitsui are enough). Also, many thanks for spotting in your comment what are the relevant L-functions in this problem! Regards. $\endgroup$ – Tom90 Feb 5 '18 at 10:10

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