{
  "abstract": "Let $p_1, \\dotsc, p_k$ be primes not exceeding $x$ ($k \\geqslant 2$), and define the additive Mertens sum \\[ S_k(x) = \\sum_{p_1 \\leqslant x} \\cdots \\sum_{p_k \\leqslant x} \\frac{1}{p_1 + \\dotsm + p_k}. \\] In contrast to Tenenbaum's generalized (multiplicative) Mertens sum, whose leading term has order $(\\log \\log x)^k$, the sum $S_k(x)$ has leading term of order $x^{k-1}/\\log^k x$. We establish the complete asymptotic expansion \\[ S_k(x) = \\frac{x^{k-1}}{\\log^k x} \\sum_{n=0}^{N} \\frac{E_{k,n}}{\\log^n x} + O_{k,N}\\left(\\frac{x^{k-1}}{\\log^{k+N+1} x}\\right) \\quad (\\forall\\, N \\geqslant 0), \\] where the coefficients are given by absolutely convergent multiple logarithmic integrals \\[ E_{k,n} = (-1)^n \\int_{(0,1]^k} \\frac{h_n(\\log t_1, \\dotsc, \\log t_k)}{t_1 + \\dotsm + t_k}\\, \\mathrm{d}\\mathbf{t}, \\] with $h_n$ the complete homogeneous symmetric polynomial of degree $n$. We derive closed-form expressions for the first two coefficients $E_{k,0}$ and $E_{k,1}$ for all $k$, and obtain the closed form for the diagonal part of the third coefficient $E_{k,2}$ (with $E_{k,2}$ fully explicit for $k \\leqslant 7$), together with a uniform closed form for the coefficients of the new constants $G(-1/m)$, where $G(z)=\\int_0^z \\mathrm{Li}_2(t)/(1+t)\\,\\mathrm dt$, whose genuinely new instances first occur for $k \\geqslant 5$ (Proposition 5.4); consequently, the first three terms of the expansions of $S_2(x)$ and $S_3(x)$ are fully explicit. For $k = 2$, we further obtain a closed-form expression for the entire sequence $\\{E_{2,n}\\}_{n \\geqslant 0}$, whose values are explicit $\\mathbb{Q}$-linear combinations of $\\log 2$ and zeta values $ζ(j)$. The proofs rely on the real-variable form of the prime number theorem, variable rescaling, and multivariate Taylor remainder estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04366",
  "author_total": 2,
  "authors": [
    {
      "name": "Daoyi Peng"
    },
    {
      "name": "Hao Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2apnhnj9vf253.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04366",
  "fulltext_url": null,
  "id": "2apnhnj9vf253",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04366",
  "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": "W7167634827",
  "pdf_url": "https://arxiv.org/pdf/2607.04366",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2apnhnj9vf253.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2607.04366",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04366"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$",
  "title_text": "Complete Asymptotic Expansion of the Additive Mertens Sum S_k(x)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/2apnhnj9vf253",
  "venue": "arXiv",
  "versions": []
}