Recently I was thinking about some questions concerning $\mathbb{Z}[x]$ and realized that they might be a bit easier if I knew the relative densities of reducible polynomials.

Let $P_d$ denote the set of all elements of $\mathbb{Z}[x]$ of degree $\leq d$. My basic question is:

Question:What fraction of elements of $P_d$ factor in $\mathbb{Z}[x]$?

The fraction $f_d$ of elements of $P_d$ which factor satisfies $f_d\geq 1-\zeta(d+1)^{-1}$ (with $\zeta$ the Riemann Zeta Function) since this is the count of elements which factor as a constant times an element of $P_d$. When $d = 0,1$ one obviously has equality, but what is not clear to me is whether $f_d = 1-\zeta(d+1)^{-1}$ holds for all $d$.

I thought about the analogous problem in $\mathbb{F}_p[x]$ (where one ignores constant factors), but in this case as $d\rightarrow\infty$ one has $f_d\rightarrow 1$. If this behavior carries over to $\mathbb{Z}[x]$ then as $d$ grows large, almost every polynomial of degree $d$ factors which seems absurd; therefore looking at this question in $\mathbb{F}_p[x]$ doesn't seem to help much.

So, letting $f_d = 1-\zeta(d+1)^{-1}+\varepsilon(d)$, I am wondering if the correction term $\varepsilon$ equals zero or, if not, what is $\varepsilon(d)$ and how would one derive it?

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