{
  "id": "1301.4530",
  "version": "v1",
  "published": "2013-01-19T04:24:55.000Z",
  "updated": "2013-01-19T04:24:55.000Z",
  "title": "Conditional Results for a Class of Arithmetic Functions: a variant of H. L. Montgomery and R. C. Vaughan's method",
  "authors": [
    "Xiaodong Cao",
    "Wenguang Zhai"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $a, b,c $ and $k$ be positive integers such that $1\\leq a\\leq b,a<c<2(a+b), c\\ne b$ and $(a,b,c)=1$. Define the arithmetic function $f_k(a,b;c;n)$ by $$ \\sum_{n=1}^{\\infty}\\frac{f_k(a,b;c;n)}{n^s}=\\frac{\\zeta (as)\\zeta (bs)}{\\zeta^k(cs)}, \\Re s >1.$$ Let $\\Delta_k(a,b;c;x)$ denote the error term of the summatory function of the function $f_k(a,b;c;n).$ IN this paper we shall give two expressions of $\\Delta_k(a,b;c;x)$. As applications, we study the so-called $(l,r)$-integers, the generalized square-full integers, the $e-r$-free integers, the divisor problem over $r$-free integers, the $e$-square-free integers. An important tool is a generalization of a method of H. L. Montgomery and R. C. Vaughan.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-19T04:24:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic function",
      "vaughans method",
      "conditional results",
      "montgomery",
      "error term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.4530C"
    }
  }
}