{
  "abstract": "For a positive integer \\(n\\), let \\(M_n=\\min_{x\\in\\mathbb R}\\sum_{k=1}^n |\\cos(kx)|\\). We determine \\(M_n\\) exactly. Apart from the exceptional values \\(M_2=1/\\sqrt2\\), \\(M_4=1+\\sqrt3/2\\), and \\(M_6=(-1+3\\sqrt5+2\\sqrt{5+2\\sqrt5})/4\\), one has \\(M_n=\\lfloor n/2\\rfloor\\). The minimizers are also classified: for \\(n\\notin\\{2,4,6\\}\\), equality is attained only at \\(x\\equiv π/2\\pmodπ\\), while the exceptional cases are attained at \\(x\\equiv\\pmπ/4\\), \\(\\pmπ/6\\), and \\(\\pmπ/10\\pmodπ\\), respectively. The proof is elementary and uses piecewise concavity, a permutation modulo \\(2q\\), and two finite trigonometric estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05545",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruixi Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9cq4aa4aq8sh1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05545",
  "fulltext_url": null,
  "id": "9cq4aa4aq8sh1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05545",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212055836",
  "pdf_url": "https://arxiv.org/pdf/2609.05545",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9cq4aa4aq8sh1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.05545",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05545"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Exact Minima of Finite Absolute Cosine Sums",
  "title_text": "Exact Minima of Finite Absolute Cosine Sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/9cq4aa4aq8sh1",
  "venue": "arXiv",
  "versions": []
}