{
  "abstract": "We study the landscape function on a discretized interval. The discrete landscape has an explicit closed form, and its spectral coefficients can be computed exactly. We show that these coefficients lie in explicit abelian extensions of $\\mathbb{Q}$ and obtain the bound $$[\\mathbb{Q}(c_k(N)):\\mathbb{Q}]\\leqφ(N)$$ for every odd $k$. The Chandra--Jain conjecture, asserting that $[\\mathbb{Q}(c_1(N)):\\mathbb{Q}]=φ(N)$ for all $N\\geq3$, has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814). We then reduce higher modes to the coprime case. Exact computations for all 624 pairs with $3\\leq N\\leq50$ reveal only the degrees $φ(N/d)$ and $2φ(N/d)$ for non-coprime modes, where $d=\\gcd(N,k)$. This motivates the Second Chandra--Jain Conjecture that this degree dichotomy holds generally. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.11372",
  "author_total": 2,
  "authors": [
    {
      "name": "Aryaman Chandra"
    },
    {
      "name": "S. R. Jain"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0m7w32q94rn2d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.11372",
  "fulltext_url": null,
  "id": "0m7w32q94rn2d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.11372",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7202350565",
  "pdf_url": "https://arxiv.org/pdf/2608.11372",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0m7w32q94rn2d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2608.11372",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.11372"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On The Spectral Properties of Discrete Landscape Functions",
  "title_text": "On The Spectral Properties of Discrete Landscape Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/0m7w32q94rn2d",
  "venue": "arXiv",
  "versions": []
}