{
  "id": "1302.0625",
  "version": "v3",
  "published": "2013-02-04T09:28:38.000Z",
  "updated": "2014-05-22T05:58:24.000Z",
  "title": "Prime polynomials in short intervals and in arithmetic progressions",
  "authors": [
    "Efrat Bank",
    "Lior Bary-Soroker",
    "Lior Rosenzweig"
  ],
  "comment": "Changes in the introduction, accepted to Duke",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p<x that are congruent to a modulo d, for d^(1+delta)<x, is about pi(x)/phi(d). More precisely, we prove: Let 1\\leq m<k be integers, let q be a prime power, and let f be a monic polynomial of degree k with coefficients in the finite field with q elements. Then there is a constant c(k) such that the number N of prime polynomials g=f+h with deg h \\leq m satisfies |N-q^(m+1)/k|\\leq c(k)q^(m+1/2). Here we assume m\\geq 2 if \\gcd(q,k(k-1))>1 and m\\geq 3 if q is even and deg f' \\leq 1. We show that this estimation fails in the neglected cases. Let \\pi_q(k) be the number of monic prime polynomials of degree k with coefficients in the finite field with q elements \\FF_q. For relatively prime polynomials f,D\\in \\FF_q[t] we prove that the number N' of monic prime polynomials g that are congruent to f modulo D and of degree k satisfies |N'-\\pi_q(k)/\\phi(D)|\\leq c(k)\\pi_q(k)q^{-1/2}/\\phi(D), as long as 1\\leq \\deg D\\leq k-3 (or \\leq k-4 if p=2 and (f/D)' is constant). We also generalize these results to other factorization types.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-22T05:58:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06"
    ],
    "keywords": [
      "arithmetic progressions",
      "short intervals",
      "monic prime polynomials",
      "establish function field versions",
      "factorization types"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.0625B"
    }
  }
}