{
  "abstract": "For $x,y>0$ let $S(x,y)$ denote the largest value of $\\sum_{n\\in A}τ(n)$ over subsets $A$ of $[1,x]\\cap\\N$ whose elements are pairwise separated by more than $y$ (here $τ$ is the divisor function). We determine the order of magnitude, and in a subrange the precise asymptotic, of $S(x,y)$ when the spacing parameter $y$ ranges over a polylogarithmic scale $y=(\\log x)^{1+λ(\\logλ-1)}$, $λ>1$. We show that for every fixed $λ>2$ the spacing constraint costs a computable power of $\\log x$ in the exponent, while for every fixed $1<λ<2$ it costs nothing at all: $S(x,y)\\sim x\\log x$, the same asymptotic as the unconstrained sum $\\sum_{n\\le x}τ(n)$. Near the threshold $λ=2$ the two regimes meet, and we show that for any fixed $κ>0,$ $S(x,y)\\asymp_κ x\\log x$ uniformly as long as $λ\\le 2+2κ/\\sqrt{\\log\\log x}$. The proof combines the Sathe-Selberg distribution of the number of prime factors, a smooth squarefree estimate obtained from a generating function together with Cauchy's formula, a sieve estimate for correlations among the squarefree parts of the integers that contribute most heavily to $\\sum_{n\\le x}τ(n)$, and two combinatorial devices (the Caro-Wei bound and a concentration inequality on the symmetric group) used to extract a well-spaced independent set from a sparse auxiliary graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31832",
  "author_total": 1,
  "authors": [
    {
      "name": "Aahan Chatterjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dnyvcspjztmhq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31832",
  "fulltext_url": null,
  "id": "dnyvcspjztmhq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31832",
  "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": "W7223687569",
  "pdf_url": "https://arxiv.org/pdf/2610.31832",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dnyvcspjztmhq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.31832",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31832"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Divisor Sums with Spacing Constraints",
  "title_text": "Divisor Sums with Spacing Constraints",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/dnyvcspjztmhq",
  "venue": "arXiv",
  "versions": []
}