{
  "id": "1710.08914",
  "version": "v1",
  "published": "2017-10-24T17:57:43.000Z",
  "updated": "2017-10-24T17:57:43.000Z",
  "title": "Primes represented by positive definite binary quadratic forms",
  "authors": [
    "Asif Zaman"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\\pi_f(x)$ within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH), and for almost all discriminants. The key feature of these estimates is that they hold whenever $x$ exceeds a small power of $D$ and, in some cases, this range of $x$ is essentially best possible. In particular, if $f$ is reduced then this optimal range of $x$ is achieved for almost all discriminants or by assuming GRH. We also exhibit an upper bound for the number of primes represented by $f$ in a short interval and a lower bound for the number of small integers represented by $f$ which have few prime factors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-24T17:57:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E16",
      "11N05"
    ],
    "keywords": [
      "positive definite binary quadratic forms",
      "positive definite integral binary",
      "definite integral binary quadratic form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}