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:



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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?



Density of Irreducible Polynomials in $\mathbb{Z}[x]$
I restore the following in clarified form as an over-sized comment; in a temporary (at least I hope it was only temporary) state of confusion I posted it as an answer, which it is not (I was not careful regarding the different notions of ir/reducibility, sorry about that).


The density of integral polynomials of fixed degree that are reducible as *rational* polynomials, i.e. that are the product of two *non-constant* integral polynomials, is $0$.
(This contains the statement for monic ones as a special case.)


More precisley, the order of the numer of such polynomials of height at most $t$ is $t^d$ for $d \ge 3$.


That is, it is known that for $d \ge 3$ there exists a constant $C\_d$ such that if $R\_d(t)$ denotes the number of reducible polynomials with height at most $t$ and degree $d$ then 
$$ t^d \le |R\_d(t)| \le C\_d t^d $$


For $d= 2$ one has an additional logarithmic factor; the order in this case is $ t^2 log t $.


This and related ressults are proved by G. Kuba in 
'On the distribution of reducible polynomials, Math. Slovaca 59 (2009), no. 3, 349–356.'


(The novelty of the paper is the quality of the estimate; the density result is much older, an upper bound of the form $t^d (log t)^2$ seems to be mentioned in Polya and Szego.)


