{
  "id": "1606.04877",
  "version": "v1",
  "published": "2016-06-15T17:37:23.000Z",
  "updated": "2016-06-15T17:37:23.000Z",
  "title": "Chebyshev's bias for products of $k$ primes",
  "authors": [
    "Xianchang Meng"
  ],
  "comment": "28 pages, 1 figure, 2 tables",
  "categories": [
    "math.NT"
  ],
  "abstract": "For any $k\\geq 1$, we study the distribution of the difference between the number of integers $n\\leq x$ with $\\Omega(n)=k$ or $\\omega(n)=k$ in two different arithmetic progressions and determine the bias between them, where $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity and $\\omega(n)$ is the number of distinct prime factors of $n$. Under some reasonable assumptions, Rubinstein and Sarnak studied this problem for primes in arithmetic progressions (i.e. $\\Omega(n)=1$); Ford and Sneed recently proved similar results for products of two primes with $\\Omega(n)=2$ in arithmetic progressions. Under the same assumptions, we solve the Chebyshev's bias problem for products of $k$ primes for all $k\\geq 1$ and for both cases of $\\Omega(n)=k$ and $\\omega(n)=k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-15T17:37:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11N60"
    ],
    "keywords": [
      "arithmetic progressions",
      "distinct prime factors",
      "chebyshevs bias problem",
      "similar results",
      "difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}