{
  "id": "2312.00722",
  "version": "v1",
  "published": "2023-12-01T16:58:30.000Z",
  "updated": "2023-12-01T16:58:30.000Z",
  "title": "Convolution identities for divisor sums and modular forms",
  "authors": [
    "Ksenia Fedosova",
    "Kim Klinger-Logan",
    "Danylo Radchenko"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove exact identities for convolution sums of divisor functions of the form $\\sum_{n_1 \\in \\mathbb{Z} \\smallsetminus \\{0,n\\}}\\varphi(n_1,n-n_1)\\sigma_{2m_1}(n_1)\\sigma_{2m_2}(n-n_1)$ where $\\varphi(n_1,n_2)$ is a Laurent polynomial with logarithms for which the sum is absolutely convergent. Such identities are motivated by computations in string theory and prove and generalize a conjecture of Chester, Green, Pufu, Wang, and Wen from \\cite{CGPWW}. Originally, it was suspected that such sums, suitably extended to $n_1\\in\\{0,n\\}$ should vanish, but in this paper we find that in general they give Fourier coefficients of holomorphic cusp forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-01T16:58:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divisor sums",
      "modular forms",
      "convolution identities",
      "holomorphic cusp forms",
      "divisor functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}