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 $n=39$:
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[![D100][1]][1]
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And it continues to diminish ...
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[![D1000000][2]][2]
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... 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?

  [1]: https://i.sstatic.net/t11Ds.jpg
  [2]: https://i.sstatic.net/DHXs0.jpg