{
  "id": "1109.3107",
  "version": "v1",
  "published": "2011-09-14T15:16:26.000Z",
  "updated": "2011-09-14T15:16:26.000Z",
  "title": "Sign Changes of the Liouville function on quadratics",
  "authors": [
    "Peter Borwein",
    "Stephen K. K. Choi",
    "Himadri Ganguli"
  ],
  "journal": "10.4153/CMB-2011-166-9",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \\vspace{1mm} \\noindent {\\bf Conjecture (Chowla).} {\\em \\begin{equation} \\label{a.1} \\sum_{n\\le x} \\lambda (f(n)) =o(x) \\end{equation} for any polynomial $f(x)$ with integer coefficients which is not of form $bg(x)^2$. } \\vspace{1mm} \\noindent The prime number theorem is equivalent to \\eqref{a.1} when $f(x)=x$. Chowla's conjecture is proved for linear functions but for the degree greater than 1, the conjecture seems to be extremely hard and still remains wide open. One can consider a weaker form of Chowla's conjecture, namely, \\vspace{1mm} \\noindent {\\bf Conjecture 1 (Cassaigne, et al).} {\\em If $f(x) \\in \\Z [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\\in \\Z[x]$, then $\\lambda (f(n))$ changes sign infinitely often.} Clearly, Chowla's conjecture implies Conjecture 1. Although it is weaker, Conjecture 1 is still wide open for polynomials of degree $>1$. In this article, we study Conjecture 1 for the quadratic polynomials. One of our main theorems is {\\bf Theorem 1.} {\\em Let $f(x) = ax^2+bx +c $ with $a>0$ and $l$ be a positive integer such that $al$ is not a perfect square. Then if the equation $f(n)=lm^2 $ has one solution $(n_0,m_0) \\in \\Z^2$, then it has infinitely many positive solutions $(n,m) \\in \\N^2$.} As a direct consequence of Theorem 1, we prove some partial results of Conjecture 1 for quadratic polynomials are also proved by using Theorem 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-14T15:16:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville function",
      "sign changes",
      "prime number theorem",
      "chowlas conjecture implies conjecture",
      "quadratic polynomials"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.3107B"
    }
  }
}