Edited after mistake in the first version. 


It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a set of asymptotic density one (Selberg's result is actually more precise: one can take $x+O(f(c) \log^2 x)$ where $f(x)$ is any function that tends to $x+\infty$ with $x$).
Here a set of density one is a subset $S \in \mathbb R\_+$ such that $\mu(S \cap [0,y])/y \longrightarrow 1$ when $y \longrightarrow \infty$. 


I have heard that this result has been generalized for primes in arithmetic progressions under GRH. I would like to know if more generally that result has been generalized for primes in a Frobenian set? More precisely,



> 
> Given a fixed Galois number field $L/\mathbb Q$, $G=Gal(L/\mathbb Q)$, $C$ a conjugacy class in $G$, is it true that for every $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ such that $Frob\_{p,Gal(L/\mathbb Q)} \in C$ for every $x$ in a set
>  of asymptotic density one (under GRH, and Artin's conjecture if you wish)? 
> 
> 
> 


I am looking for any reference discussing that question... If there is none available
(as it seems at first glance according to my looking in mathscinet), I would be also interested to the clearest references you know treating the case of arithmetic sequence. 
In any case, thanks... 



Primes in short intervals with a preassigned frobenius
One reference where a Hoheisel type result (right number of such 
primes in every interval $(x,x+x^{1-\delta})$ for some $\delta>0$) is proved unconditionally is the paper by Balog and Ono "The Chebotarev density theorem and 
some questions of Serre" (see <http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/062.pdf>). Selberg's method is very robust (all it uses 
is that the number of zeros in intervals of length $1$ is bounded by some 
constant times log(conductor)), and it would be a simple 
matter to take the explicit formula and bound the variance of primes 
in short intervals on GRH. 


