The Bateman--Horn Conjecture predicts how often an irreducible polynomial  assumes prime values. We will discuss how with sufficient averaging in the coefficients of the polynomial (exponential in the size of the inputs), one can not only prove Bateman--Horn results on average but also pin down precise information about the distribution of prime values at finite but growing scales. We will prove that 100% of polynomials satisfy the appropriate analogue of the Poisson Tail Conjecture, in the sense that the distribution of the gaps between consecutive prime values around the average spacing is Poisson.

We will also study the frequencies of sign patterns of the Liouville function evaluated at the consecutive outputs of f; viewing f as a random variable, we establish the limiting distribution for every sign pattern. 
A key input behind all of our arguments is Leng's recent quantitative work on the higher-order Fourier uniformity of the von Mangoldt and M\"obius functions (in turn relying on Leng, Sah, and Sawhney's quantitative inverse theorem for the Gowers norms).

This talk is based on joint work with Noah Kravitz and Max Xu.