{
  "abstract": "Friedlander and Iwaniec proved that the number of points of the orbit $\\{γi \\colon γ\\in\\SL_2(\\Z)\\}$ lying at a distance $p-2$ from the origin $i$ of the upper half-plane, with $p\\le x$ prime, is of order $x/\\log x$; the upper bound is unconditional, while the lower one rests on a strong hypothesis concerning the distribution of primes in arithmetic progressions. We consider instead square-free distances and prove unconditionally an asymptotic formula, whose main term is of order $x$. Employing the weighted linear sieve, we also show unconditionally that the square-free distances $n$ with at most $7$ prime factors contribute $\\gg x/\\log x$. Finally, assuming that the sequence $r(n-2)r(n+2)$ has level of distribution $x^θ$ in arithmetic progressions, where $r(n)$ is the number of ways to write $n$ as a sum of two squares, we obtain the same lower bound for the distances with at most $N$ prime factors (the value $7$ corresponds to $θ=1/6$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28200",
  "author_total": 1,
  "authors": [
    {
      "name": "Alisa Sedunova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/de4xv1kcvh8yd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28200",
  "fulltext_url": null,
  "id": "de4xv1kcvh8yd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28200",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214244175",
  "pdf_url": "https://arxiv.org/pdf/2609.28200",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/de4xv1kcvh8yd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28200",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28200"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the hyperbolic prime number theorem",
  "title_text": "On the hyperbolic prime number theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/de4xv1kcvh8yd",
  "venue": "arXiv",
  "versions": []
}