{
  "id": "1704.07979",
  "version": "v1",
  "published": "2017-04-26T05:56:30.000Z",
  "updated": "2017-04-26T05:56:30.000Z",
  "title": "Unexpected biases in prime factorizations and Liouville functions for arithmetic progressions",
  "authors": [
    "Snehal M. Shekatkar",
    "Tian An Wong"
  ],
  "comment": "25 pages, 6 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of P\\'olya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-26T05:56:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A51",
      "11N13",
      "11N37",
      "11F66"
    ],
    "keywords": [
      "prime factorization",
      "arithmetic progression",
      "unexpected biases",
      "classical liouville function",
      "polyas conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}