(Sorry if this is a naive question; it is not my area!)
Consider the following strengthening of Artin's conjecture on primitive roots (and Dirichlet's theorem) for the case of $n=2$: every arithmetic sequence $a + b \mathbf{Z}$ with $\gcd(a,b)=1$ and $\gcd(a,2)=1$ contains infinitely many primes $p$ such that $2$ is a primitive root mod $p$.
As a particular case, are there infinitely many primes $p \equiv 1 \pmod{4}$ such that $2$ is primitive mod $p$?
Does this follow either from Artin's conjecture (or its quantitative version) or the GRH (like Artin's conjecture does), or something stronger?
For example, the list A001122 of $p$ such that $2$ is a primitive root mod $p$ contains, for every odd prime $q \leq 19$ and every $1 \leq b < q$, an entry $p$ such that $p \equiv b \pmod{q}$. This at least a priori makes it possible to imagine that it contains infinitely many such entries (assuming Artin's conjecture, of course!). Also, without the condition $\gcd(a,2)=1$ (i.e. $a$ is odd), then the strengthening fails, of course, as $2$ is a primitive root mod $p$ only if $p \equiv 3,5 \pmod{8}$ (otherwise it is a quadratic residue).
Best Answer
It's plausible that there are infinitely many primes $p \equiv 1 \bmod 4$ of which $2$ is a primitive residue. However, this is false for $p \equiv \pm 1 \bmod 8$, because $2$ is a quadratic residue. It seems reasonable to guess that this is the only obstruction.
Likewise (mutatis mutandis) for other fixed $r$ being a primitive root modulo infinitely many $p$ in an arithmetic progression: as long as $r$ satisfies the necessary conditions that $r$ is neither $-1$ nor a square, the only excluded arithmetic progressions should be ones in which Quadratic Reciprocity makes $r$ a square $\bmod p$.
edited to add: The OP's comment reminds me that Artin's conjecture, though still open, is known to be a consequence of the GRH (generalized Riemann hypothesis) for zeta functions of number fields; Wikipedia helpfully provides a reference (Hooley [1]), and further reminds that Heath-Brown [2] proved unconditionally that Artin's conjecture holds for all primes $r$ with at most two exceptions. I'd expect that these results extend with little further effort to $p$ in arithmetic progressions in which $r$ is not always a square $\bmod p$; possibly such extensions are already in the literature.
References
[1] Hooley, Christopher (1967). On Artin's conjecture. J. reine angew. Math. 225, 209--220. doi:10.1515/crll.1967.225.209
[2] Heath-Brown, D.R. (March 1986): Artin's Conjecture for Primitive Roots. The Quarterly Journal of Mathematics 37 (1), 27--38. doi:10.1093/qmath/37.1.27