It is well known that for a given polynomial $f \in \mathbb{Z}[x]$ the number of primes $p$ s.t. $f$ has a root modulo $p$ is infinite. In fact, one can even write down a formula for the density of such primes (e.g. D. Berend, Y. Bilu "Polynomials with roots modulo every integer").

Now suppose that $f = \sum_{i=0}^d a_ix^i$ is irreducible, the degree $d$ is fixed and all $a_i \leq N^C$ for some constant $C$. Is it possible to prove the uniform bound (e.g. whith the constant depending only on $d$ and $C$ and $N$ large) $$ \pi(N, f) = \Omega_{d, C}(\frac{N}{\log N}), $$ where $\pi(N, f)$ means the number of primes less than $N$ s.t. $f$ has a root modulo $p$?

The aforementioned paper cites a general density formula due to Lagarias and Odlyzko which seem to provide such a bound assuming GRH. It's of course highly desirable to drop this assumption.