Procounting measures and the Bateman--Horn conjecture

Let $D$ be the ring of $S$-integers in a global field and $\text{da}$ its profinite completion. We propose a profinite version of the Bateman–Horn conjecture over $D$ and provide a first comparison with the classical one and its generalizations. Our approach is based on the new notion of procounting measure: a distribution on $\text{da}$ which should be seen as a profinite analogue of the counting function for a subset of $\mathbb{R}$. This allows us to deal with subsets of $\text{da}$ having Haar measure $0$ (corresponding to density zero in $\mathbb{R}$).