The profinite Bateman–Horn conjecture over S-integers

Let $D$ be the ring of $S$-integers in a global field, with characteristic different from $2$, and let $\widehat D$ be its profinite completion. Write $\mathcal P(D)$ for the prime elements of $D$, $\widehat D^*$ for the group of units of $\widehat D$, and let $f_1,\dots,f_k\in D[x]$ be non-associate separable irreducible polynomials with product $\mathbf f$. Assume

$$f(\widehat D)\cap\widehat D^*\ne\varnothing.$$

Using the normalized counting measures and their limiting procounting measures on the relevant residue rings, one has

Profinite Bateman–Horn conjecture. The following limits exist and satisfy

$$\lim_{\mathfrak n\to0}\mu_{\mathbf f|_D^{-1}(\mathcal P(D)^k),\mathfrak n}=\mu_{\mathbf f^{-1}((\widehat D^*)^k)},$$

and

$$\lim_{\mathfrak n\to0}\mu_{\mathbf f(D)\cap\mathcal P(D)^k,\mathfrak n}=\mu_{\mathbf f(\widehat D)\cap(\widehat D^*)^k}.$$

The second equality is called the direct-image version.

This is a profinite, measure-theoretic analogue of Bateman–Horn: local unit conditions encode the absence of congruence obstructions, while procounting measures retain information invisible to Haar measure. The conjecture is proposed in the source and is not known in this generality.