In praise of the Bateman-Horn conjecture

Joint work with Gareth Jones, with computational assistance from Jean Bétréma
Let there be a set of polynomials $f_1,\ldots,f_k\in\mathbb{Z}[t]$. We are interested in the situation when all the values $f_1(t),\ldots,f_k(t)$ are simultaneously prime. The question to which the Bateman-Horn conjecture gives an answer is: for a given $x$, how many $t\le x$ are there for which the above situation occurs.
For example:
(1) when there is only one polynomial $f_1(t)=t$, the answer is given by the Prime Number Theorem (Hadamard and de la Vallée Poussin, 1896);
(2) when there is still one polynomial $f_1(t)=at+b$, Dirichlet (1837) proved that there are infinitely many $t\in\mathbb{N}$ such that $f_1(t)$ is prime; the number of $t\le x$ was established later;
(3) when there are two polynomials $f_1(t)=t$ and $f_2(t)=t+2$, we have the Twin Primes conjecture;
(4) a question which is also in this framework: are there infinitely many projective groups of prime degree?
The Bateman-Horn conjecture predicts, with an astonishing accuracy, the number of the "good" values of $t$.
The talk will be given in a mixed Russian-Western style. Namely:
(1) the slides will be in English;
(2) the talk itself will be in English;
(3) but the duration of the talk may turn out to be significantly longer than a polite one hour.