{
  "abstract": "We derive explicit formulas for finite sums of powers of cotangents, both at rational and arbitrary arguments, together with their alternating counterparts, weighted by additive characters. Our method relies on Apostol--Bernoulli polynomials and on the partial fraction expansion of trigonometric functions. As applications, we solve the problem posed by Berndt, Kim and Zaharescu \\cite{BKZ:2023} on evaluating $\\sum\\csc^{2m}$ and $\\sum\\sec^{2m}$ at rational arguments, recover as a special case the classical Berndt--Yeap formula, and relate our identities to spectral zeta functions of circulant graphs. Furthermore, to systematically evaluate higher-order derivatives arising from the Lipschitz summation formula, we introduce a novel sequence of polynomial coefficients $c_{m,j}(β)$, termed \\emph{deformed cotangent coefficients}, and establish their recursive and algebraic properties. Using these coefficients, we establish a general additive-and-multiplicatively twisted character mean value theorem for Lerch zeta functions, significantly extending the framework of Bayad and Raouj to incorporate additive character twists.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01801",
  "author_total": 2,
  "authors": [
    {
      "name": "Abdelmejid Bayad"
    },
    {
      "name": "Salah Eddine Rihane"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5jvs7bp1pfjk9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01801",
  "fulltext_url": null,
  "id": "5jvs7bp1pfjk9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01801",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01801",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5jvs7bp1pfjk9.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01801",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01801"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finite Cotangent Power Sums Weighted by Additive Characters With Application to Mean Values of Additive-and-Multiplicatively Twisted $L$-Functions",
  "title_text": "Finite Cotangent Power Sums Weighted by Additive Characters With Application to Mean Values of Additive-and-Multiplicatively Twisted L-Functions",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/5jvs7bp1pfjk9",
  "venue": "arXiv",
  "versions": []
}