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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The distribution functions of $ \sigma(n)/n$ and $ n/\varphi(n)$


Author: Andreas Weingartner
Journal: Proc. Amer. Math. Soc. 135 (2007), 2677-2681
MSC (2000): Primary 11N25, 11N60
Published electronically: February 6, 2007
MathSciNet review: 2317939
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Abstract: Let $ \sigma(n)$ be the sum of the positive divisors of $ n$. We show that the natural density of the set of integers $ n$ 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.


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Additional Information

Andreas Weingartner
Affiliation: Department of Mathematics, Southern Utah University, Cedar City, Utah 84720
Email: weingartner@suu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08771-0
PII: S 0002-9939(07)08771-0
Keywords: Natural density, sum-of-divisors function, Euler's totient function
Received by editor(s): April 13, 2006
Received by editor(s) in revised form: May 4, 2006
Published electronically: February 6, 2007
Communicated by: Ken Ono
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.