{
  "abstract": "Let $p(n)$ denote the least prime factor of $n$ and $L_C(x)=Cx^{1/2}(\\log x)^2$ with $C>0$. The sum of $p(n)/n$ over composite $n$ lying in the short interval $[x,\\,x+L_C(x)]$, a question raised by Erdős and Graham, is studied. (i) The constant $c=8$ in the mean asymptotic is estimated \\[ S(x)=\\sum_{n<x,\\ n\\ \\text{composite}}\\frac{p(n)}{n}=\\frac{c\\,x^{1/2}}{(\\log x)^2}\\Bigl(1+O\\Bigl(\\frac1{\\log x}\\Bigr)\\Bigr) +O\\Bigl(\\frac{x^{1/3}}{\\log x}\\Bigr). \\] (ii) For every fixed $C>0$, the window sums $μ_C(x):=\\sum_{x\\le n\\le x+L_C(x)}p(n)/n$ over composites have mean $4C$: $\\frac1X\\sum_{x\\le X}μ_C(x)=4C+O_C(1/\\log X)$, and second moment $\\frac1X\\sum_{x\\le X}(μ_C(x)-4C)^2=O_C(1)$. (iii)Under the Maier-consistent bounded-fluctuation hypothesis, $μ_C(x)=4C+O_C(1)$ holds uniformly in $x$, and the Erdős--Graham question has an affirmative answer. Unconditionally, the uniform statement remains open; proving it would require resolving short-interval prime estimates at scale $(\\log y)^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24930",
  "author_total": 1,
  "authors": [
    {
      "name": "Xu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86dv8ev2tmyx7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24930",
  "fulltext_url": null,
  "id": "86dv8ev2tmyx7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24930",
  "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": "W7204468165",
  "pdf_url": "https://arxiv.org/pdf/2608.24930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86dv8ev2tmyx7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2608.24930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24930"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the sum of least prime factors in short intervals",
  "title_text": "On the sum of least prime factors in short intervals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/86dv8ev2tmyx7",
  "venue": "arXiv",
  "versions": []
}