Probability that polynomial $f(m)$ is prime for $m < n$, as $n$ goes to infinity?

Question

Say we have a polynomial $f(n)$ with integer coefficients, and we want to know the probability that $f(m)$ is prime for $m<n$, as $n$ goes to infinity. The prime number theorem treats the case $f(n) = n$. Has any work been done on this question for more general polynomials, for example quadratics?

Answer 1

There are no proven results of strength comparable to the Prime Number Theorem, except for degree $1$ polynomials which are covered by PNT-AP. For instance, we don't know a single polynomial of degree $>1$ which provably takes infinitely many prime values, even though we believe this should be true as long as the polynomial is irreducible and doesn't have any obvious reason to not be prime (i.e. being divisible by a fixed prime). But there is a heuristic expectation for what the correct probability should be, which is given precisely by the Bateman-Horn conjecture:

https://en.wikipedia.org/wiki/Bateman%2DHorn_conjecture

For linear polynomials, the situation is pretty well understood. If $(a,b)>1$, then the polynomial $an+b$ obviously takes only finitely many prime values (at most one, in fact), so the probability is either $0$ or $1/n$. Otherwise, $a$ and $b$ are relatively prime, and the asymptotic probability of $an+b$ being prime is

$$P(am+b \text{ prime} \mid m \le n) \sim \frac{a}{\phi(a) \log n}.$$

(Here $a$ and $b$ are assumed to be fixed so the rate of convergence is allowed to depend on $a$ and $b$.)