# Polynomials dense with primes

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?

• For degree $1$ and co-prime coefficients $\Delta>0$ (Dirichlet's theorem, Chebotarev density theorem). For degree >1 (and any irreducible $p$), it is an open problem . – user6976 Apr 21 '18 at 12:56
• @MarkSapir That would be true if he were multiplying the density by $\log n$, but he isn't, so in fact every polynomial has density zero. – Will Sawin Apr 21 '18 at 13:59
• @WillSawin: Yes, of course. But I thought that log was just missing. Clearly one needs to divide by the number of primes $\le n$ and not by $n$. – user6976 Apr 21 '18 at 19:18
• Possible duplicate of Bateman-Horn conjecture, continued – Felipe Voloch Apr 22 '18 at 8:58

It is known (it follows from Brun's sieve, or more modern sieves) that for any fixed polynomial $p$, there exists a constant $c_p$ such that $$\# \{ n\le x\colon p(n) \text{ is prime} \} < c_p \frac x{\log x}.$$ In particular, your density $\Delta$ equals $0$ for any polynomial (as Will Sawin commented).

For irreducible polynomials without obvious obstructions (such as all the values being even), it is conjectured that $\# \{ n\le x\colon p(n) \text{ is prime} \} \sim s_p \frac x{\log x}$ for some constant $s_p$ as $x\to\infty$; but this is an open problem for any polynomial $p$ of degree greater than $1$.

This has been asked multiple times before (with three variations by yours truly), so is a mega-duplicate, if you will:

Bateman-Horn, continued even further

Bateman-Horn conjecture, continued

Unexpectedly prime rich cubic polynomial

And even resulted in a preprint: