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

1 Answer 1

up vote 2 down vote accepted

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

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