{
  "id": "1802.00085",
  "version": "v1",
  "published": "2018-01-31T22:13:20.000Z",
  "updated": "2018-01-31T22:13:20.000Z",
  "title": "Explicit bounds for primes in arithmetic progressions",
  "authors": [
    "Michael A. Bennett",
    "Greg Martin",
    "Kevin O'Bryant",
    "Andrew Rechnitzer"
  ],
  "comment": "66 pages. Results of computations, and the code used for those computations, can be found at: http://www.nt.math.ubc.ca/BeMaObRe/",
  "categories": [
    "math.NT"
  ],
  "abstract": "We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\\gcd(a,q)=1$ and $3 \\leq q \\leq 10^5$, and $\\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p \\equiv a \\pmod{q}$ with $p \\leq x$, we show that $$ \\bigg| \\theta (x; q, a) - \\frac{x}{\\phi (q)} \\bigg| < \\frac1{3600} \\frac q{\\phi(q)} \\frac{x}{\\log x}, $$ for all $x \\geq 7.94 \\cdot 10^9$ (with sharper constants obtained for individual such moduli $q$). We establish inequalities of the same shape for the other standard prime-counting functions $\\pi(x;q,a)$ and $\\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\\pmod q$ when $q\\le4500$. For moduli $q>10^5$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,\\chi)$ for quadratic characters $\\chi$, and an improved explicit upper bound for exceptional zeros.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-31T22:13:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N13",
      "11N37",
      "11M20",
      "11M26",
      "11Y35",
      "11Y40"
    ],
    "keywords": [
      "arithmetic progressions",
      "explicit bounds",
      "explicit lower bound",
      "stronger explicit inequalities",
      "derive explicit upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}