{
  "id": "2302.05165",
  "version": "v1",
  "published": "2023-02-10T10:45:33.000Z",
  "updated": "2023-02-10T10:45:33.000Z",
  "title": "The distribution of the multiplicative index of algebraic numbers over residue classes",
  "authors": [
    "Pieter Moree",
    "Antonella Perucca",
    "Pietro Sgobba"
  ],
  "comment": "15 pages, comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\\times$. Given a prime $\\mathfrak p$ of $K$ we denote by ${\\rm ind}_{\\mathfrak p}(G)$ the index of the subgroup $(G\\bmod\\mathfrak p)$ of the multiplicative group of the residue field at $\\mathfrak p$. Under the Generalized Riemann Hypothesis we determine the natural density of primes of $K$ for which this index is in a prescribed set $S$ and has prescribed Frobenius in a finite Galois extension $F$ of $K$. We study in detail the natural density in case $S$ is an arithmetic progression, in particular its positivity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-10T10:45:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R45",
      "11A07",
      "11R44"
    ],
    "keywords": [
      "algebraic numbers",
      "residue classes",
      "multiplicative index",
      "distribution",
      "natural density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}