{
  "abstract": "Let $K$ be a field containing $\\mathbb{Q}$, let $Γ$ be the ring of all functions from the positive integers to $K$ under Dirichlet convolution, and let $Γ_n$ be its truncation to functions supported on $[1,n]$. In [Snellman, Homology Homotopy Appl. 2 (2000), 17-27; arXiv:math/9904143] it was shown that $Γ_n$ is a polynomial ring modulo a monomial ideal $I_n$ which is stable after reversing the order of the variables, and the Poincare-Betti series of $Γ_n$ was computed in terms of the numbers $C_{n,v}$ of minimal generators of $I_n$ of least support $v$. We prove that $C_{n,v} = Φ(n,p_v)$, Legendre's sifting function: the number of integers in $[1,n]$ free of prime factors $\\le p_v$. This identifies an invariant of a minimal free resolution with a classical object of sieve theory, and it refines to the grading by total degree, with the number of prime factors carried in a variable. As consequences we obtain: a proof of Conjecture 4.6 of the 2000 paper, which was left open there; the asymptotics $C_n \\sim π(n)^2/2$ of the total number of minimal generators, showing that the lower bound $C_n \\ge \\binom{π(n)+1}{2}$ of that paper is asymptotically sharp, together with the exact order $C_n - \\binom{π(n)+1}{2} \\sim \\frac{16}{3} n^{3/2} / \\log^3 n$ of the error; and the identification of $C_n$ with the OEIS sequence A182843. We also record errata for the 2000 paper: one stated result is false, one worked example is miscomputed, and two proofs are incomplete. Corrected statements and complete proofs are given.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29555",
  "author_total": 1,
  "authors": [
    {
      "name": "Jan Snellman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3n25xew4ncn31.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29555",
  "fulltext_url": null,
  "id": "3n25xew4ncn31",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29555",
  "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.AC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.29555",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3n25xew4ncn31.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2608.29555",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29555"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Truncations of the ring of number-theoretic functions, revisited",
  "title_text": "Truncations of the ring of number-theoretic functions, revisited",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/3n25xew4ncn31",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.3384/lith-mat-r-2026-03",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}