Year: 2026
Type: preprint
Abstract: Let $D$ be the ring of $S$-integers in a global field and $\widehat{D}$ 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 $\widehat{D}$ 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 $\widehat{D}$ having Haar measure $0$ (corresponding to density zero in $\mathbb{R}$).
Source: arXiv (Cornell University)
Institution: Xi’an Jiaotong-Liverpool University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166683456