TITLE: Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial
QUESTION [7 upvotes]: The starting point for this question is the following (false) statement


$\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$


Given a polynomial function $p:\mathbb{N} \to \mathbb{N}$ we define $$\text{prime}(p) = \{n\in\mathbb{N}: p(n) \text{ is prime}\}.$$
For $A\subseteq \mathbb{N}$ we define the upper density by $$\operatorname{ud}(A)=\limsup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
Is there a non-constant polynomial $p$ such that $\operatorname{ud}(\text{prime}(p)) > 0$? 
If yes, what is $\sup\{\operatorname{ud}(\text{prime}(p)): p:\mathbb{N}\to\mathbb{N} \text{ is a non-constant polynomial}\}$?

REPLY [8 votes]: This is a supplement to Eric Naslund excellent answer, and also an elaboration of Terry Tao's comment above. The crucial asymptotic formula
$$\frac{1}{\pi(x)}\sum_{p\leq x}v_{F}(p)=1+o(1)$$
also follows from two 19th century results: the Frobenius theorem on splitting types and the Cauchy-Frobenius orbit counting lemma (also known as Burnside's lemma). 
Indeed, consider the Galois group $G$ of the splitting field of $F$ over $\mathbb{Q}$ as a permutation group acting on the roots of $F$. The Frobenius theorem implies that the left hand side is asymtotically the average number of fixed points of $G$, which by the Cauchy-Frobenius formula equals the number of orbits of $G$. However, $G$ acts transitively, so the number of orbits equals $1$.
Added 1. As Vesselin Dimitrov pointed out, the above asymptotic formula also follows from Landau's prime ideal theorem (published in 1903).
Added 2. I just learned from the paper Stevenhagen-Lenstra: Chebotarev and his Density Theorem that the above asymptotic formula was originally shown by Kronecker (published in 1880), and this formed the basis of the quoted work of Frobenius of more group theoretic flavor.