{
  "abstract": "We give $L^2$-bounds with respect to the height of the spectral parameter for Epstein zeta functions $\\mathrm{Z}_Q(s)$ and non-holomorphic Eisenstein series $E_{\\mathfrak{c}}(s,z)$ attached to the Hecke congruence subgroup $Γ_0 (p)$ of prime level $p$. Our bounds are completely explicit in the level and, additionally, constitute an almost sure logarithmic improvement over previously established bounds for the level-one Eisenstein series. Following the approach of Kelmer--Kontorovich--Lutsko, we combine the approximate functional equation for $\\mathrm{Z}_Q(s)$ with estimates for the number of pairs of lattice vectors at which $Q$ takes nearly equal values. Under a Diophantine hypothesis on $Q$, we sharpen these counting estimates using results of Eskin--Margulis--Mozes on the distribution of values of quadratic forms of signature $(2,2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37420",
  "author_total": 2,
  "authors": [
    {
      "name": "Jitendra Bajpai"
    },
    {
      "name": "Kristian Holm"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ch9ebxxm0p3sq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37420",
  "fulltext_url": null,
  "id": "ch9ebxxm0p3sq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37420",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7224121311",
  "pdf_url": "https://arxiv.org/pdf/2610.37420",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ch9ebxxm0p3sq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37420",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37420"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Spectral $L^2$-bounds on Eisenstein series and Diophantine properties of quadratic forms",
  "title_text": "Spectral L^2-bounds on Eisenstein series and Diophantine properties of quadratic forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/ch9ebxxm0p3sq",
  "venue": "arXiv",
  "versions": []
}