{
  "id": "1607.01882",
  "version": "v1",
  "published": "2016-07-07T06:45:56.000Z",
  "updated": "2016-07-07T06:45:56.000Z",
  "title": "Large bias for integers with prime factors in arithmetic progressions",
  "authors": [
    "Xianchang Meng"
  ],
  "comment": "12 pages. Comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the distribution of the product of $k ~(\\geq 2)$ distinct primes $p_1\\cdots p_k\\leq x$ with each prime factor in an arithmetic progression $p_j\\equiv a_j \\bmod q$, $(a_j, q)=1$ $(q \\geq 3, 1\\leq j\\leq k)$. For any $A>0$, we prove an asymptotic formula uniformly for $2\\leq k\\leq A\\log\\log x$ for the number of such integers. Moreover, we show that, there are large biases towards some certain arithmetic progressions $\\boldsymbol{a}:=(a_1, \\cdots, a_k)$, and such biases have connections with Mertens theorem and the least prime in arithmetic progressions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-07T06:45:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11N13",
      "11N69"
    ],
    "keywords": [
      "arithmetic progression",
      "prime factor",
      "large bias",
      "asymptotic formula",
      "distinct primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}