{
  "abstract": "We evaluate the characteristic polynomial of a random unitary matrix at the points on the unit circle where the polynomial takes real values. In analogy to the Riemann zeta function we call these the ``random matrix Gram points''. On the random matrix side we answer in the affirmative a question of Richard Hall, who expected that zeta on the Gram points should be larger than typical. We prove that for any fixed positive $k$, the $2k$th moment of the characteristic polynomial averaged over the random matrix Gram points is larger than the $2k$th moment of the characteristic polynomial averaged over the whole unit circle. For large $N$ the leading-order asymptotic term is the same size, but all lower-order terms are larger. In an appendix we show how to adapt the heuristic recipe of Conrey--Farmer--Keating--Rubinstein--Snaith to conjecture moments of zeta at Gram points, and show by example that the second and fourth moments calculated by following the recipe agree with previously proven results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39856",
  "author_total": 2,
  "authors": [
    {
      "name": "Christopher Hughes"
    },
    {
      "name": "Somin Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ezde9psjfmn78.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39856",
  "fulltext_url": null,
  "id": "ezde9psjfmn78",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39856",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39856",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ezde9psjfmn78.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.39856",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39856"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Moments of characteristic polynomials at random matrix Gram points",
  "title_text": "Moments of characteristic polynomials at random matrix Gram points",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/ezde9psjfmn78",
  "venue": "arXiv",
  "versions": []
}