{
  "id": "2306.10425",
  "version": "v1",
  "published": "2023-06-17T21:11:46.000Z",
  "updated": "2023-06-17T21:11:46.000Z",
  "title": "Murmurations and explicit formulas",
  "authors": [
    "Alex Cowan"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Unexpected oscillations in $a_p$ values in a family of elliptic curves were observed experimentally by He, Lee, Oliver, and Pozdnyakov. We propose a heuristic explanation for these oscillations based on the \"explicit formula\" from analytic number theory. A crucial ingredient in this heuristic is that the distribution of the zeros of the associated $L$-functions has a quasi-periodic structure. We present empirical results for both a family of elliptic curves as well as a family of quadratic Dirichlet characters whose values exhibit similar oscillations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-17T21:11:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "explicit formula",
      "murmurations",
      "elliptic curves",
      "quadratic dirichlet characters",
      "analytic number theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}