{
  "abstract": "We investigate the sums ∑ n ≤ X , ( n , q ) = 1 μ ( n ) n s log k ⁡ ( X n ) \\sum _{n\\le X, (n,q)=1}\\frac {\\mu (n)}{n^{s}}\\log ^{k}\\left (\\frac {X}{n}\\right ) , where k ∈ { 0 , 1 } k\\in \\{0,1\\} , s ∈ C s\\in \\mathbb {C} , ℜ s > 0 \\Re s>0 . Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality ∑ n ≤ X Λ ( n ) / n ≤ log ⁡ X \\sum _{n\\le X}\\Lambda (n)/n\\le \\log X , valid for any X ≥ 1 X\\geq 1 .",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Olivier Ramaré",
      "orcid": "0000-0002-8765-0465"
    },
    {
      "name": "Sebastian Zuniga Alterman",
      "orcid": "0000-0003-2555-5339"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2sdv5m27y1y1h.bib",
  "cited_by": 5,
  "doi": "10.1090/mcom/4216",
  "fulltext_url": null,
  "id": "2sdv5m27y1y1h",
  "issns": [
    "0025-5718",
    "1088-6842"
  ],
  "landing_url": "https://doi.org/10.1090/mcom/4216",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W4389600268",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-23",
  "publisher": "American Mathematical Society",
  "ris": "https://research.draconitic.com/paper/2sdv5m27y1y1h.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T13:21:00Z",
      "native_id": "W4389600268",
      "source": "openalex",
      "url": "https://doi.org/10.1090/mcom/4216"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Möbius function and primes: an identity factory with applications",
  "title_text": "Möbius function and primes: an identity factory with applications",
  "type": "peer-reviewed",
  "updated_at": "2026-11-06T13:20:22Z",
  "url": "https://research.draconitic.com/paper/2sdv5m27y1y1h",
  "venue": "Mathematics of Computation",
  "versions": []
}