{
  "abstract": "We study several probabilistic questions concerning the digits of $s(n)$, the sum of proper divisors of an integer $n$. In particular, we show that $s(n)$ obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function $k(x) \\rightarrow \\infty$, almost all integers $n \\leq x$ have every decimal digit occurring among the first $k(x)$ digits and the last $k(x)$ digits of $s(n)$. We also present an upper bound for the number of composite integers $n$ up to $x$ for which $s(n)$ is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs $n$ were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set $s^{-1}(\\mathcal{A})$, where $\\mathcal{A}$ is a set of integers with missing digits. Our result for composite $n$ shows that the count is much smaller when prime inputs are excluded.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18981",
  "author_total": 5,
  "authors": [
    {
      "name": "Kübra Benli"
    },
    {
      "name": "Cécile Dartyge"
    },
    {
      "name": "Charlotte Dombrowsky"
    },
    {
      "name": "Paul Pollack"
    },
    {
      "name": "Lola Thompson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9b1e67h5pt0nt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18981",
  "fulltext_url": null,
  "id": "9b1e67h5pt0nt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18981",
  "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": "W7170055178",
  "pdf_url": "https://arxiv.org/pdf/2607.18981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9b1e67h5pt0nt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2607.18981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the digits of the sum of proper divisors",
  "title_text": "On the digits of the sum of proper divisors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/9b1e67h5pt0nt",
  "venue": "arXiv",
  "versions": []
}