Let $p(n)$ be a polynomial with integer coefficients.
Define $\Delta( p(n) )$, the *prime density* of $p(n)$, to be
the limit of the ratio with respect to $n$
of the number of primes $p(k)$ generated when the polynomial
is evaluated at the natural numbers $k=1,2,\ldots,n$:
$$
\Delta( p(n) ) \;=\; \lim_{n \to \infty}
\frac{ \textrm{number of } p(k), k \le n, \textrm{that are prime}}
{n}
$$
For example, Euler's polynomial $p(n)=n^2+n+41$
starts out with ratio $1$, but then diminishes
beyond $n=39$:

And it continues to diminish ...

... and by $n=10^7$ has reached $\Delta=0.22$.

Q. What is the largest known $\Delta( p(n) )$ over all polynomials $p(n)$?

In particular, are there any polynomials known to have $\Delta > 0$?

Maybe these questions can be answered assuming one or more conjectures?