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Procounting measures and the Bateman--Horn conjecture

  • Luca Demangos
  • Ignazio Longhi
  • Francesco Maria Saettone
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Abstract

Let D be the ring of S-integers in a global field and D̂ 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 D̂ which should be seen as a profinite analogue of the counting function for a subset of ℝ. This allows us to deal with subsets of D̂ having Haar measure $0$ (corresponding to density zero in ℝ).

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Cite

BibTeXRIS
BibTeX
@misc{demangos2026procounting,
  title = {{Procounting measures and the Bateman--Horn conjecture}},
  author = {Luca Demangos and Ignazio Longhi and Francesco Maria Saettone},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.29250},
  eprint = {2606.29250},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.29250},
}
RIS
TY  - UNPB
TI  - Procounting measures and the Bateman--Horn conjecture
AU  - Luca Demangos
AU  - Ignazio Longhi
AU  - Francesco Maria Saettone
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2606.29250
UR  - https://arxiv.org/abs/2606.29250
AB  - Let D be the ring of S-integers in a global field and D̂ 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 D̂ which should be seen as a profinite analogue of the counting function for a subset of ℝ. This allows us to deal with subsets of D̂ having Haar measure $0$ (corresponding to density zero in ℝ).
ER  -