{
  "abstract": "For each integer k >= 1, let sigma_k(n) = sum_{d|n} d^k, let sigma_k^{-1} denote its Dirichlet inverse, and let J_k denote the kth Jordan totient function. Using established prime-power values of sigma_k^{-1}, we first obtain the elementary criterion sigma_k^{-1}(n) + J_k(n) + 2 = 0 precisely when n is prime. At k = 1, this involves Euler's totient J_1 = phi and provides a prime-only comparison for the next equation. Our main result classifies the zeros of sigma_k^{-1}(n) + mu(n) + J_k(n) + 3: for k = 1, they are precisely the primes and 18; for k >= 2, they are precisely the primes. The proof separates integers according to their prime-exponent patterns. Sign and divisibility arguments exclude composite zeros outside the cube-free nonsquarefree case, where an additional divisibility condition isolates 18 at k = 1 and precludes composite zeros for higher k. Standard convolution and Lambert-series identities provide the arithmetic framework, while an intermediate-divisor relation explains the exceptional zero. These criteria are structural characterizations rather than efficient primality tests, since direct evaluation of the multiplicative formulas generally presupposes a factorization of n.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33278",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhichen Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vdnbj3cdkf1t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33278",
  "fulltext_url": null,
  "id": "7vdnbj3cdkf1t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33278",
  "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": "W7214699606",
  "pdf_url": "https://arxiv.org/pdf/2609.33278",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vdnbj3cdkf1t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.33278",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33278"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Prime-Detecting Identities from Dirichlet Inversion",
  "title_text": "Prime-Detecting Identities from Dirichlet Inversion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/7vdnbj3cdkf1t",
  "venue": "arXiv",
  "versions": []
}