{
  "id": "1305.3972",
  "version": "v2",
  "published": "2013-05-17T03:19:59.000Z",
  "updated": "2025-05-10T20:54:16.000Z",
  "title": "Multiplicity one for $L$-functions and applications",
  "authors": [
    "David W. Farmer",
    "Ameya Pitale",
    "Nathan C. Ryan",
    "Ralf Schmidt"
  ],
  "comment": "Significant revision. Provides more motivation behind our approach and includes different applications",
  "categories": [
    "math.NT"
  ],
  "abstract": "We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to $L$-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for $L$-functions. Significantly stronger results are available when the L-function is known to be automorphic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-17T03:19:59.000Z",
      "abstract": "We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients satisfy a partial Ramanujan bound and do not differ by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to $L$-functions. These results follow from our main result about $L$-functions.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2025-05-10T20:54:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiplicity",
      "applications",
      "partial ramanujan bound",
      "main result",
      "number-theoretic objects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.3972F"
    }
  }
}