The distribution functions of $\sigma (n)/n$ and $n/\varphi (n)$
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- by Andreas Weingartner PDF
- Proc. Amer. Math. Soc. 135 (2007), 2677-2681 Request permission
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.References
- H. Davenport, Ăśber numeri abundantes, Preuss. Akad. Wiss. Sitzungsber. (1933), 830-837.
- Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. 7 (1998), no. 2, 137–143. MR 1677091, DOI 10.1080/10586458.1998.10504363
- P. Erdös, Some remarks about additive and multiplicative functions, Bull. Amer. Math. Soc. 52 (1946), 527–537. MR 16078, DOI 10.1090/S0002-9904-1946-08604-8
- A. I. Vinogradov, On the remainder in Merten’s formula, Dokl. Akad. Nauk SSSR 148 (1963), 262–263 (Russian). MR 0143740
- Isac Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z. 28 (1928), no. 1, 171–199 (German). MR 1544950, DOI 10.1007/BF01181156
- G. Tenenbaum, V. Toulmonde, Sur le comportement local de la répartition de l’indicatrice d’Euler, Funct. Approx. Comment. Math., to appear.
- Vincent Toulmonde, Module de continuité de la fonction de répartition de $\phi (n)/n$, Acta Arith. 121 (2006), no. 4, 367–402 (French). MR 2224402, DOI 10.4064/aa121-4-6
Additional Information
- Andreas Weingartner
- Affiliation: Department of Mathematics, Southern Utah University, Cedar City, Utah 84720
- MR Author ID: 678374
- Email: weingartner@suu.edu
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2677-2681
- MSC (2000): Primary 11N25, 11N60
- DOI: https://doi.org/10.1090/S0002-9939-07-08771-0
- MathSciNet review: 2317939