{
  "id": "1811.01613",
  "version": "v1",
  "published": "2018-11-05T11:09:36.000Z",
  "updated": "2018-11-05T11:09:36.000Z",
  "title": "On the zeros of Epstein zeta functions near the critical line",
  "authors": [
    "Yoonbok Lee"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $Q$ be a positive definite quadratic form with integral coefficients and let $E(s,Q)$ be the Epstein zeta function associated with $Q$. Assume that the class number of $Q$ is bigger than $1$. Then we estimate the number of zeros of $E(s,Q)$ in the region $ \\Re s > \\sigma_T ( \\theta ) := 1/2 + ( \\log T)^{- \\theta}$ and $ T < \\Im s < 2T$, to provide its asymptotic formula for fixed $ 0 < \\theta < 1$ conditionally. Moreover, it is unconditional if the class number of $Q$ is $2$ or $3$ and $ 0 < \\theta < 1/13$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-05T11:09:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "epstein zeta function",
      "critical line",
      "class number",
      "positive definite quadratic form",
      "asymptotic formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}