{
  "abstract": "We study the zeros of the quasimodular forms $G_{\\{2\\}^n}$ defined by iterated sums. We first show that, for every $n>0$, $G_{\\{2\\}^n}$ has exactly $n$ simple zeros on each of the vertical half-lines $\\Real(τ)=0$ and $\\Real(τ)=1/2$, and that the zeros for consecutive values of $n$ satisfy an interlacing property. The proof is based on an expression of $G_{\\{2\\}^n}$ in terms of the $n$-th derivative of $η^3$ and on the theory of bell-shaped functions, rather than on Rankin--Swinnerton-Dyer method. We also determine the asymptotic behavior of these zeros as $n\\to\\infty$. In addition, we prove that all zeros of $G_{\\{2\\}^n}$ are simple and that $G_{\\{2\\}^n}$ has infinitely many $SL_2(\\ZZ)$-inequivalent zeros. We further show that quasimodular forms of maximal depth have no zeros at CM points. In particular, none of the zeros of $G_{\\{2\\}^n}$ are CM points. Finally, in the special case $G_{2,2}$, we show that each Ford circle contains exactly two distinct simple zeros.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12729",
  "author_total": 2,
  "authors": [
    {
      "name": "Katsumi Kina"
    },
    {
      "name": "Gyucheol Shin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mbjg907nwb86.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12729",
  "fulltext_url": null,
  "id": "9mbjg907nwb86",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12729",
  "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|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212566677",
  "pdf_url": "https://arxiv.org/pdf/2609.12729",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mbjg907nwb86.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.12729",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12729"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Zeros of Quasimodular Forms Defined by Iterated Sums",
  "title_text": "Zeros of Quasimodular Forms Defined by Iterated Sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/9mbjg907nwb86",
  "venue": "arXiv",
  "versions": []
}