The distribution functions of <IMG BORDER="0" SRC="/proc/2007-135-09/S0002-9939-07-08771-0/gif-title0/img1.gif" > and <IMG BORDER="0" SRC="/proc/2007-135-09/S0002-9939-07-08771-0/gif-title0/img2.gif" >期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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The distribution functions of and

Authors:	Andreas Weingartner

Institution:	Department of Mathematics, Southern Utah University, Cedar City, Utah 84720

Abstract:	Let $\sigma(n)$ be the sum of the positive divisors of . We show that the natural density of the set of integers satisfying $\sigma(n)/n\ge t$ is given by $\exp\left\{ -e^{t \, e^{-\gamma}} \left(1+O\left({t^{-2}}\right)\right) \right\}$ , where $\gamma$ denotes Euler's constant. The same result holds when $\sigma(n)/n$ is replaced by $n/\varphi(n)$ , where $\varphi$ is Euler's totient function.

Keywords:	Natural density sum-of-divisors function Euler's totient function

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