{
  "id": "2210.02294",
  "version": "v1",
  "published": "2022-10-05T14:32:59.000Z",
  "updated": "2022-10-05T14:32:59.000Z",
  "title": "Infinitely many zeros of additively twisted $L$-functions on the critical line",
  "authors": [
    "Doyon Kim"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For $f$ a cuspidal modular form for the group $\\Gamma_0(N)$ of integral or half-integral weight, $N$ a multiple of $4$ in case the weight is half-integral, we study the zeros of the $L$-function attached to $f$ twisted by an additive character $e^{2\\pi i n \\frac{p}{q}}$ with $\\frac{p}{q}\\in \\mathbb{Q}$. We prove that for certain $f$ and $\\frac{p}{q}\\in \\mathbb{Q}$, the additively twisted $L$-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses distributions to prove the result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-05T14:32:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41",
      "11F37"
    ],
    "keywords": [
      "critical line",
      "cuspidal modular form",
      "half-integral weight",
      "hardy-littlewood method",
      "additive character"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}