{
  "abstract": "We study a level analog of Euler's totient function for integer mosaics. Let $P_i(n)$ be the set of primes appearing in the first $i$ levels of the mosaic of $n$, and let $φ_i(n)$ count the integers $m\\leq n$ for which $P_i(m)\\cap P_i(n)=\\varnothing$. We prove a Möbius divisor-sum formula for $φ_i(n)$ and reduce it to a sum over a set $\\mathcal{V}_{i,S}$ of powerful integers. If $S\\neq\\varnothing$, $i\\geq 2$, and $q=\\min S$, then \\[ |\\mathcal{V}_{i,S}\\cap[1,x]|\\sim C_{i,S}x^{1/q}, \\] with $C_{i,S}$ an explicit positive Euler-product constant. For fixed $S$, the density $δ_i(S)$ of integers whose first $i$ levels avoid $S$ exists and has an Euler product; for nonempty $S$, $i\\ge2$, and $q=\\min S$, the number of such integers up to $N$ is $δ_i(S)N+O_{i,S}(N^{1/q})$. Taking $S=P_i(n)$ gives \\[ \\frac{φ_i(n)}{n}=δ_i(P_i(n))+O_\\varepsilon(n^{-1/2+\\varepsilon}) \\] uniformly in $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.22842",
  "author_total": 3,
  "authors": [
    {
      "name": "The Anh Dao"
    },
    {
      "name": "Tim Kasian"
    },
    {
      "name": "Mostafa Mirabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chgkgwqjatdpw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.22842",
  "fulltext_url": null,
  "id": "chgkgwqjatdpw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.22842",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7165668553",
  "pdf_url": "https://arxiv.org/pdf/2606.22842",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chgkgwqjatdpw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2606.22842",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.22842"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Level totients for integer mosaics",
  "title_text": "Level totients for integer mosaics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/chgkgwqjatdpw",
  "venue": "arXiv",
  "versions": []
}