Rocky Mountain Journal of Mathematics

On the distributions of $\sigma(n)/n$ and $n/\varphi(n)$

Emre Alkan

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 43, Number 3 (2013), 713-736.

Dates
First available in Project Euclid: 1 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1375361971

Digital Object Identifier
doi:10.1216/RMJ-2013-43-3-713

Mathematical Reviews number (MathSciNet)
MR3093262

Zentralblatt MATH identifier
1368.11105

Subjects
Primary: 11N25: Distribution of integers with specified multiplicative constraints 11N60: Distribution functions associated with additive and positive multiplicative functions

Keywords
Distributions density sum of divisors function Euler's totient function arithmetic progressions l -free integers

Citation

Alkan, Emre. On the distributions of $\sigma(n)/n$ and $n/\varphi(n)$. Rocky Mountain J. Math. 43 (2013), no. 3, 713--736. doi:10.1216/RMJ-2013-43-3-713. https://projecteuclid.org/euclid.rmjm/1375361971


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References

  • E. Alkan, K. Ford and A. Zaharescu, Diophantine approximation with arithmetic functions I, Trans. Amer. Math. Soc. 361 (2009), 2263-2275.
  • –––, Diophantine approximation with arithmetic functions II, Bull. Lond. Math. Soc. 41 (2009), 676-682.
  • F. Behrend, Three reviews; of papers by Chowla, Davenport and Erdös, Jahr. Fort. Math. 60 (1935), 146-149.
  • S. Chowla, On abundant numbers, J. Indian Math. Soc. 1 (1934), 41-44.
  • H. Davenport, Über numeri abundantes, Sitz. Akad. Wiss. Berlin 27 (1933), 830-837.
  • –––, Multiplicative number theory, Third edition, Grad. Text. Math. 74, Springer-Verlag, New York, 2000.
  • P. Erdös, On the density of abundant numbers, J. London Math. Soc. 9 (1934), 278-282.
  • –––, Some remarks about additive and multiplicative functions, Bull. Amer. Math. Soc. 42 (1946), 527-537.
  • –––, On the distribution of numbers of the form $\sigma(n)/n$ and some related questions, Pacific J. Math. 52 (1974), 59-65.
  • P. Erdös and A. Wintner, Additive arithmetical functions and statistical independence, Amer. J. Math. 61 (1939), 713-721.
  • A. Languasco and A. Zaccagnini, A note on Mertens' formula for arithmetic progressions, J. Number Theory 127 (2007), 37-46.
  • K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin, 1957.
  • I. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z. 28 (1928), 171-199.
  • E.C. Titchmarsh, The theory of the Riemann zeta-function, Second edition, Oxford University Press, New York, 1986.
  • V. Toulmonde, Module de continuité de la fonction de répartition de $\varphi(n)/n$, Acta Arith. 121 (2006), 367-402.
  • –––, Sur les variations de la fonction de répartition de $\varphi(n)/n$, J. Number Theory 120 (2006), 1-12.
  • A.I. Vinogradov, On Mertens' theorem, Dokl. Akad. Nauk. SSSR 143 (1962), 1020-1021.
  • –––, On the remainder in Mertens' formula, Dokl. Akad. Nauk. SSSR 148 (1963), 262-263.
  • A. Weingartner, The distribution functions of $\sigma(n)/n$ and $n/\varphi(n)$, Proc. Amer. Math. Soc. 135 (2007), 2677-2681. \noindentstyle