{
  "abstract": "Let $d(n)$ denote the number of divisors of a positive integer $n$. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum $\\sum_{n \\leq x} \\frac{1}{d(n)}$, with Ramanujan having introduced an asymptotic formula for this sum with an explicit evaluation for the constant $A_1$ for the leading term $A_1 \\frac{x}{\\sqrt{\\log x}}$. Gabdullin et al. recently considered a hybrid of this problem and the Titchmarsh divisor problem concerning $\\sum_{p\\leq x} d(p-1)$, proving that $\\sum_{p\\leq x} \\frac{1}{d(p-1)} \\asymp \\frac{x}{(\\log x)^{3/2}}$. This result and Erdős's asymptotic formula $ \\sum_{n \\leq x} d(d(n)) \\sim c_1 \\, x \\log \\log x $ for a constant $c_1 \\in (0, \\infty)$ lead us to consider the hybrid $\\sum_{n \\leq x} \\frac{1}{d(d(n))}$ of the Erdős and Ramanujan divisor sums. In this paper, we prove that there exists a constant $c_3 \\in (0, \\infty)$ such that $$\\sum_{n \\leq x} \\frac{1}{d(d(n))} \\sim c_3 \\frac{x}{ \\log \\log x}, $$ through a combined application of Golomb's estimate for powerful numbers, Turán's quantitative form of the Hardy-Ramanujan theorem, and a fourth-moment estimate for the number of distinct prime divisors due to Granville and Soundararajan. Moreover, an explicit formula for the constant $c_3$ is given.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.00695",
  "author_total": 1,
  "authors": [
    {
      "name": "John M. Campbell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/brzv44t1watrv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.00695",
  "fulltext_url": null,
  "id": "brzv44t1watrv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.00695",
  "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": "W7160114121",
  "pdf_url": "https://arxiv.org/pdf/2605.00695",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/brzv44t1watrv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2605.00695",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.00695"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On divisor sums due to Erdős and Ramanujan",
  "title_text": "On divisor sums due to Erdős and Ramanujan",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/brzv44t1watrv",
  "venue": "arXiv",
  "versions": []
}