{
  "id": "1204.6689",
  "version": "v12",
  "published": "2012-04-30T16:26:38.000Z",
  "updated": "2015-12-23T15:07:05.000Z",
  "title": "On a pair of zeta functions",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "22 pages. Accepted version for publication in Int. J. Number Theory",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $m$ be a positive integer, and define $$\\zeta_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}{n^s}\\ \\ \\text{and} \\ \\ \\zeta^*_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}{n^s},$$ for $\\Re(s)>1$, where $\\omega(n)$ denotes the number of distinct prime factors of $n$, and $\\Omega(n)$ represents the total number of prime factors of $n$ (counted with multiplicity). In this paper we study these two zeta functions and related arithmetical functions. We show that $$\\sum^\\infty_{n=1\\atop n\\ \\text{is squarefree}}\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n=0\\quad\\text{if}\\ \\ m>4,$$ which is similar to the known identity $\\sum_{n=1}^\\infty\\mu(n)/n=0$ equivalent to the Prime Number Theorem. For $m>4$, we prove that $$\\zeta_m(1):=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n=0 \\ \\ \\text{and}\\ \\ \\zeta^*_m(1):=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}n=0,$$ and that both $V_m(x)(\\log x)^{2\\pi i/m}$ and $V_m^*(x)(\\log x)^{2\\pi i/m}$ have explicit given finite limits as $x\\to\\infty$, where $$V_m(x)=\\sum_{n\\le x}\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n\\ \\ \\text{and}\\ \\ V_m^*(x)=\\sum_{n\\le x}\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}n.$$ We also raise a hypothesis on the parities of $\\Omega(n)-n$ which implies the Riemann Hypothesis.",
  "revisions": [
    {
      "version": "v11",
      "updated": "2014-05-19T01:11:04.000Z",
      "abstract": "Let $m$ be a positive integer, and define $$\\zeta_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}{n^s}\\ \\ \\text{and} \\ \\ \\zeta^*_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}{n^s},$$ for $\\Re(s)>1$, where $\\omega(n)$ denotes the number of distinct factors of $n$, and $\\Omega(n)$ represents the total number of prime factors of $n$ (counted with multiplicity). In this paper we study these two zeta functions and related arithmetical functions. We show that $$\\sum^\\infty_{n=1\\atop n\\ \\text{is squarefree}}\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n=0\\quad\\text{if}\\ \\ m>4,$$ which is similar to the known identity $\\sum_{n=1}^\\infty\\mu(n)/n=0$ equivalent to the Prime Number Theorem. For $m>4$, we prove that $$\\zeta_m(1):=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n=0 \\ \\ \\text{and}\\ \\ \\zeta^*_m(1):=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}n=0,$$ and that both $V_m(x)(\\log x)^{2\\pi i/m}$ and $V_m^*(x)(\\log x)^{2\\pi i/m}$ have explicit given finite limits as $x\\to\\infty$, where $$V_m(x)=\\sum_{n\\le x}\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n\\ \\ \\text{and}\\ \\ V_m^*(x)=\\sum_{n\\le x}\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}n.$$ We also raise a hypothesis on the parities of $\\Omega(n)-n$ which implies the Riemann Hypothesis.",
      "comment": "22 pages, refined version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v12",
      "updated": "2015-12-23T15:07:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M99",
      "11A25",
      "11N37"
    ],
    "keywords": [
      "zeta functions",
      "prime number theorem",
      "riemann hypothesis",
      "distinct factors",
      "finite limits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.6689S"
    }
  }
}