{
  "id": "2201.10771",
  "version": "v1",
  "published": "2022-01-26T06:41:00.000Z",
  "updated": "2022-01-26T06:41:00.000Z",
  "title": "Estimates for $L$-functions in the critical strip under GRH with effective applications",
  "authors": [
    "Aleksander Simonič"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\\log{\\mathcal{L}(s)}$ and $\\mathcal{L}'(s)/\\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\\mathcal{L}(s)$ are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood's well known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application we provide conditional and effective estimate for the Mertens function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-26T06:41:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11N37"
    ],
    "keywords": [
      "effective applications",
      "critical strip",
      "explicit upper bounds",
      "polynomial euler product",
      "mertens function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}