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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...

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  • $\begingroup$ You're both right. I will reformulate my question... $\endgroup$ – Joël Jun 7 '13 at 20:54
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    $\begingroup$ I think the first result in AP was: K. Prachar, Über den Primzahlsatz von A. Selberg, Acta. Arith., 28 (1975), pp. 277–297. eudml.org/doc/205389 $\endgroup$ – v08ltu Jun 8 '13 at 3:07
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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.

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    $\begingroup$ You're good at this. You should consider a career in mathematics. $\endgroup$ – Will Jagy Aug 23 '13 at 3:56

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