{
  "id": "2008.13407",
  "version": "v1",
  "published": "2020-08-31T07:51:36.000Z",
  "updated": "2020-08-31T07:51:36.000Z",
  "title": "The fourth moment of Dirichlet $L$-functions at the central value",
  "authors": [
    "Xiaosheng Wu"
  ],
  "comment": "74 pages, any comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "The fourth moment of Dirichlet $L$-functions at the central point has been widely concerned, and an asymptotic formula with a power saving error term has been proved by Young for prime moduli, while the general moduli case is still open. In this work, we prove an asymptotic formula with a power saving error term for the fourth moment of Dirichlet $L$-functions at the central point for general moduli. The work relies on our deduction of the theory for a special divisor sum function, called $\\mathcal{D}_q$-function, which is the key to deduce the main term and to apply the Kuznetsov trace formula for congruence groups. Another key ingredient is an uniform bound for a double sum in Kloosterman sums, which is of independent interest and may be useful elsewhere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-31T07:51:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06"
    ],
    "keywords": [
      "fourth moment",
      "central value",
      "power saving error term",
      "central point",
      "asymptotic formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}