My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of integers in cosets coprime to a fixed ideal (in some reasonable subset of the canonical embedding).


In particular, for the integers, one has the pleasant identity $$\sum\_{n \leq x , n \equiv a \pmod{q}} \Lambda(n) = \frac{1}{\phi(q)} \sum\_{\chi} \bar{\chi}(a) \sum\_{n \leq x} \Lambda(n) \chi(n). $$ Then one can use classical tools from analytic number theory, such as zero-free regions for $L$-functions, to estimate the sum on the right.


For number fields, it seems like there are significant obstacles to the above approach. Is there a reference that addresses this? (the friendlier, the better!). 


Edit: Allow me to be more precise (in response to KConrad). Fix $K$ to be a number field and $O\_K$ the ring of integers. Let $I \leq O\_K$ be an ideal and $R$ be some bounded region in the canonical embedding, which we shall denote $i$. Let $a \in O\_K$ such that $((a) , I) = 1$. I am interested in $$\sum\_{\substack{\alpha \in R \\ \alpha - i(a) \in I}} 1\_P(\alpha),$$ where $1\_P(\alpha)$ is one if $(\alpha)$ is a prime ideal and zero otherwise. 



Primes in arithmetic progressions in number fields
Theorem 6 in Chapter XV, $\S$5 of Lang *Algebraic Number Theory* is a result of the sort you want. Lang formulates it using ideles, but he gives an application in more classical language in Example 3 on the following page. I first quote what he shows in Example 3, then I'll try to rewrite it to sound a bit more like what you want.


**Lang's Example 3** Let $k$ be a number field of class number $1$. Let $k\_{\infty}$ be the product of the archimedean completions of $k$, so $k\_{\infty} \cong \mathbb{R}^s \times \mathbb{C}^t$ if $k$ has $s$ real and $2t$ imaginary embeddings. Let $k\_{\infty}^{\ast}$ be the group of invertible elements of $k\_{\infty}$, so $k\_{\infty}^{\ast} \cong \mathbb{R}^{s+t} \times (\mathbb{Z}/2)^s \times (S^1)^t$, and let $k\_{\infty}^{\ast,1}$ be the subgroup of elements with norm $1$. Let $U$ be the unit group of $k$, so $k\_{\infty}^{\ast} / U \cong \mathbb{R} \times (S^1)^{s+2t-1} \times \mathbb{Z}/2^j$ for some $j$ and $k\_{\infty}^{\ast,1} / U \cong (S^1)^{s+2t-1} \times \mathbb{Z}/2^j$. Let $\sigma : k\_{\infty}^{\ast} / U \to S^1$ be a continuous homomorphism whose restriction to $k\_{\infty}^{\ast,1} / U$ is surjective. Then $\sigma(\pi)$ is equidistributed in $S^1$, as $\pi$ ranges over generators of prime ideals, each prime ideal taken once.


Okay, but you also want to be allowed to impose congruence conditions on your ideals and work with other class numbers. Lang's Theorem 6 does that, except he only states it idelically. I believe the following is the classical version. 


Let $\mathfrak{m}$ be a (nonzero) ideal of $k$ and let $H$ be the $\mathfrak{m}$-ray class group, meaning ideals relatively prime to $\mathfrak{m}$ modulo principal ideals whose generators are $1 \bmod \mathfrak{m}$. By Cebatorov, prime ideals are equidistributed in the finite group $H$, so we can consider a single class of $H$ and ask about the archimedean behavior of primes in that class. 


Fix $I$ an ideal in the desired class, so an ideal is in this class if and only if it is of the form $\alpha I$ for some $\alpha \in I^{-1}$ with $\alpha = 1 \bmod \mathfrak{m}$. Let $P \subset I^{-1}$ be the set of $\alpha$ which are $1 \bmod \mathfrak{m}$ for which $\alpha I$ is prime. We have $P \subset I^{-1} \subset k \otimes \mathbb{R} = k\_{\infty}$. Roughly, we are going to claim that $P$ is equidistributed.


Let $U$ be the group of units which are $1 \bmod \mathfrak{m}$. Choose a continuous group homomorphism $\sigma$ retracting $k^{\ast}\_{\infty}/U$ onto $k^{\ast,1}\_{\infty}/U$. Then $\sigma(P)$ is equidistributed in the compact group $k^{\ast,1}\_{\infty}/U$. 


If this is too abstract, note that a concrete case is equidistribution of Gaussian primes in wedge shaped regions of $\mathbb{C}$. See [here](https://mathoverflow.net/questions/133410/hecke-equidistribution), which is where I learned of the Lang reference.


