{
  "id": "2012.07781",
  "version": "v2",
  "published": "2020-12-14T18:19:33.000Z",
  "updated": "2021-08-04T15:07:03.000Z",
  "title": "Fourier optimization and quadratic forms",
  "authors": [
    "Andrés Chirre",
    "Oscar E. Quesada-Herrera"
  ],
  "comment": "31 pages. Minor edits. To appear in Q. J. Math",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\\ell\\geq 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\\ell$. This allows us to obtain unconditional Brun-Titchmarsh-type results in short intervals, and a conditional Cram\\'er-type result on the maximum gap between primes represented by a given positive definite quadratic form.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-08-04T15:07:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E16",
      "11M26",
      "11N05",
      "11R29",
      "11R42",
      "42B10"
    ],
    "keywords": [
      "fourier optimization",
      "positive definite quadratic form",
      "unconditional brun-titchmarsh-type results",
      "fourier analysis approach",
      "conditional cramer-type result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}