Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$.


Consider the following sum $\pi\_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$.


Can we get an upper bound for $\pi\_f(x)$?



An upper bound for the number of prime numbers in non-linear progressions
Yes, you can get sharp upper bound, matching (up to a multiplicative constant) the asymptotic predicted by the Bateman-Horn conjecture (<https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture>). This is a standard (and almost the first) application of sieve methods. I recommend the books by Halberstam and Richert, or Cojocaru and Murty. 


