{
  "abstract": "We prove that broad families of finite Dirichlet sums exhibit deterministic concentration of normalized vertical zero density on a single line. The main result gives a general criterion under which the normalized Jessen potentials converge locally uniformly to a piecewise-linear convex function with a single corner. The associated normalized measures describing the distribution of real parts of zeros in vertical mean density then converge weakly to a Dirac mass. Thus, for each fixed truncation, zeros may occupy a nontrivial range of real parts, while asymptotically their normalized vertical density concentrates on one line. The proof combines the finite Bohr lift with a translation-uniform anti-concentration estimate for isolated prime coordinates. We apply the criterion to partial sums of the Riemann zeta function, fixed Dirichlet $L$-functions, and primitive holomorphic Hecke eigenforms of fixed level, trivial Dirichlet character, and without complex multiplication. The concentration line is $\\operatorname{Re} s=1/2$ in the normalized setting and becomes $\\operatorname{Re} s=k/2$ for the classical Fourier coefficients of a form of weight $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17875",
  "author_total": 1,
  "authors": [
    {
      "name": "Eric Dubon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4nbqwdgzw46b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17875",
  "fulltext_url": null,
  "id": "b4nbqwdgzw46b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17875",
  "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": "W7213522996",
  "pdf_url": "https://arxiv.org/pdf/2609.17875",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4nbqwdgzw46b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17875",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17875"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Zero-Density Concentration for Dirichlet Polynomials",
  "title_text": "Zero-Density Concentration for Dirichlet Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/b4nbqwdgzw46b",
  "venue": "arXiv",
  "versions": []
}