Starting with $n^2+1$ and the twin primes, the talk introduces the Bateman–Horn conjecture on how often a family of polynomials simultaneously takes prime values. It explains the conjecture’s constant through local factors that adjust the prediction based on independence by counting the allowed residue classes modulo each prime.

The second part introduces the zeta distribution and formulates a question about prime values of polynomials in terms of an expectation. Drawing on Golomb’s identity and a formula of Hindry and Rivoal for the constant, the talk examines the interchange of a limit and an infinite sum that remains to be justified in this approach.