TITLE: About the prime divisors of values of polynomials
QUESTION [16 upvotes]: Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$.

Is it true that  $\sum_{p \in \mathscr P_P} \frac1 p$ diverges?

The case of polynomials with degree $2$ can be studied by Dirichlet's theorem on arithmetic progressions (I suppose a quadratic-residue argument can be applied for every case), but I am not sure how to study the cases of greater degree.  
Any reference or proof (or disproof, which, by the way, would be very surprising) would be appreciated.

REPLY [11 votes]: I don't think one needs to invoke the Chebotareev Density Theorem here. 
Assume that the given sum converges, and let $K$ be a number field generated by any root of $P(x)$. Then, by assumption, the sum of $\mathrm{Norm}(\mathfrak{p})^{-1}$ over the degree $1$ prime ideals $\mathfrak{p}$ in $K$ converges, hence the same holds for the sum of $\mathrm{Norm}(\mathfrak{p})^{-1}$ over all prime ideals $\mathfrak{p}$ in $K$. This implies that the Dedekind zeta function of $K$,
$$ \zeta_K(s)=\prod_{\mathfrak{p}}\frac{1}{1-\mathrm{Norm}(\mathfrak{p})^{-s}},\qquad \Re(s)>1, $$ 
tends to a finite limit as $s\to 1+$, while it is known that $\zeta_K(s)$ has a simple pole at $s=1$.