{
  "abstract": "We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for $\\bigl(ζ'/ζ\\bigr)^{k}$ on the critical line; we establish the connection between this problem and the short-interval variance of the convoluted von Mangoldt function $Λ^{*k}$ in short intervals and in arithmetic progressions. Under a shifted moments conjecture and the Riemann Hypothesis (RH), we prove that the short-interval variance leading coefficient is the derivative of the mean-square leading coefficient. A unitary trace-sum average governs the variance of sums of $Λ^{*k}$ over monic polynomials in $\\mathbb F_q[T]$; we denote its leading coefficient by $δ_k(c)$, where the total trace degree and matrix size grow with a fixed ratio $c$. Under a weaker shifted moments conjecture and RH, we recover this $δ_k(c)$ as the leading coefficient of the variance of short-interval sums of $Λ^{*k}$ over the integers. It is the counterpart of $γ_k(c)$, the piecewise polynomial from the variance of divisor sums, with the Dirichlet coefficients of $ζ(s)^k$ replaced by those of $\\bigl(-ζ'/ζ\\bigr)^k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05000",
  "author_total": 4,
  "authors": [
    {
      "name": "Siegfred Baluyot"
    },
    {
      "name": "John Brian Conrey"
    },
    {
      "name": "Ayesha Irfan"
    },
    {
      "name": "Nina C. Snaith"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4rg91tsjymtqa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05000",
  "fulltext_url": null,
  "id": "4rg91tsjymtqa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05000",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05000",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4rg91tsjymtqa.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.05000",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05000"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Almost primes in short intervals and averages of long Dirichlet polynomial approximations to $(ζ'/ζ)^{k}$",
  "title_text": "Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)^k",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:29Z",
  "url": "https://research.draconitic.com/paper/4rg91tsjymtqa",
  "venue": "arXiv",
  "versions": []
}