{
  "id": "1807.11093",
  "version": "v1",
  "published": "2018-07-29T18:19:58.000Z",
  "updated": "2018-07-29T18:19:58.000Z",
  "title": "Zeros of partial sums of $L$-functions",
  "authors": [
    "Arindam Roy",
    "Akshaa Vatwani"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider a certain class of multiplicative functions $f: \\mathbb N \\rightarrow \\mathbb C$. Let $F(s)= \\sum_{n=1}^\\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \\sum_{n\\le N} f(n)n^{-s}$ be the truncated Dirichlet series. In this setting, we obtain new Hal\\'asz-type results for the logarithmic mean value of $f$. More precisely, we prove estimates for the sum $\\sum_{n=1}^x f(n)/n$ in terms of the size of $|F(1+1/\\log x)|$ and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums $F_N(s)$. In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field $K$. More precisely, we give some improved results for the number of zeros up to height $T$ as well as new zero density results for the number of zeros up to height $T$, lying to the right of $\\Re(s) =\\sigma$, where $\\sigma > 1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-29T18:19:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41"
    ],
    "keywords": [
      "partial sums",
      "establish non-trivial zero-free regions",
      "zero density results",
      "mean value estimates",
      "logarithmic mean value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}