{
  "id": "1412.1157",
  "version": "v1",
  "published": "2014-12-03T01:50:07.000Z",
  "updated": "2014-12-03T01:50:07.000Z",
  "title": "Partial sums of biased random multiplicative functions",
  "authors": [
    "Marco Aymone",
    "Vladas Sidoravicius"
  ],
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "Let $\\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\\{f(p)\\}_{p\\in\\mathcal{P}}$ form a sequence of $\\pm1$ valued independent random variables with $\\mathbb{E} f(p)<0$, $\\forall p\\in \\mathcal{P}$. The function $f$ is called strongly biased (towards classical M\\\"obius function), if $\\sum_{p\\in\\mathcal{P}}\\frac{f(p)}{p}=-\\infty$ a.s., and it is weakly biased if $\\sum_{p\\in\\mathcal{P}}\\frac{f(p)}{p} $ converges a.s. Let $M_f(x):=\\sum_{n\\leq x}f(n)$. We establish a number of necessary and sufficient conditions for $M_f(x)=o(x^{1-\\alpha})$ for some $\\alpha>0$, a.s., when $f$ is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if $M_{f_\\alpha}(x)=o(x^{1/2+\\epsilon})$ for all $\\epsilon>0$ a.s., for each $\\alpha>0$, where $\\{f_\\alpha \\}_\\alpha$ is a certain family of weakly biased random multiplicative functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-03T01:50:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial sums",
      "valued independent random variables",
      "riemann hypothesis holds",
      "weakly biased random multiplicative functions",
      "squarefree integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.1157A"
    }
  }
}