Skip to main content
SpringerLink
Log in

Erdős’s Work on the Sum of Divisors Function and on Euler’s Function

  • Chapter
Book cover Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 25))

  • 2561 Accesses

Abstract

The following notation will be used throughout

log r x is the r times iterated logarithm of x,

$$\begin{gathered}\sigma _1 (n) = \sigma (n),\quad \sigma _k (n) = \sigma (\sigma _{k - 1} (n)), \hfill \\\phi _1 (n) = \phi (n),\quad \phi _k (n) = \phi (\phi _{k - 1} (n)), \hfill \\s_1 (n) = s(n) = \quad \sigma (n) - n,\quad s_i (n) = s_1 (s_{i - 1} (n)); \hfill \\\end{gathered}$$

ψ is Euler’s constant, \(\alpha _0 = \log \Pi _p \;prime\;(1 - \frac{1}{p})^{ - 1/p}\). Density is always the asymptotic density.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. T. Bateman, The distribution of values of the Euler function, Acta Arith. 21 (1972), 329–345.

    MATH  MathSciNet  Google Scholar 

  2. J. Browkin and A. Schinzel, On integers not of the form n-φ(n), Colloq. Math. 68 (1995), 55–58.

    MATH  MathSciNet  Google Scholar 

  3. H. Davenport, Über numeri abundantes, Sber. Preuß. Akad. Wiss. Berlin, 27 (1933), 830–837, also in the Collected works, vol. 4, 1834–1841.

    Google Scholar 

  4. M. Deléglise, Bounds for the density of abundant integers, Exp. Math. 7 (1998), 137–143.

    Article  MATH  Google Scholar 

  5. R. E. Dressier, A density which counts multiplicity, Pacific. J. Math. 34 (1970), 371–378.

    Article  MathSciNet  Google Scholar 

  6. P. D. T. A. Elliott, Probabilistic number theory, I Mean-value theorems, Grundlehren der Mathematischen Wissenschaften 239, Springer Verag 1979.

    Google Scholar 

  7. A. Flammenkamp and F. Luca, Infinite families of non-cototients, Colloq. Math. 86 (2000), 37–41.

    MATH  MathSciNet  Google Scholar 

  8. K. Ford, The distribution of totients, Ramanujan J. 2 (1998), 67–151.

    Article  MATH  MathSciNet  Google Scholar 

  9. -, The number of solutions of φ(x) = m, Ann. of Math. 150 (1999), 283–311.

    Article  MATH  MathSciNet  Google Scholar 

  10. -, An explicit sieve bound and small values of σ(φ(m)), Period. Mat. Hungar, 43 (2011), 15–29.

    Article  Google Scholar 

  11. K. Ford, F. Luca and C. Pomerance, Common values of the arithmetic functions φ and σ, Bull. London Math. Soc. 42 (2010), 478–488.

    Article  MATH  MathSciNet  Google Scholar 

  12. K. Ford and P. Pollack, On common values of ϕ(n) and σ(n), I, Acta Math. Hungar. 133 (2011), 251–271.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. W. Graham, J. J. Holt and C. Pomerance, On the solutions to ϕ(n) = ϕ(n +k), Number theory in Progress, vol. 2, 867–882, Walter de Gruyter 1999.

    Google Scholar 

  14. P. Loomis and F. Luca, On totient abundant numbers, Integers. Electronic J. of Combinatorial Number Theory 8 (2008), #A06.

    MathSciNet  Google Scholar 

  15. A. Ivić, The distribution of positive abundant numbers, Studia Sc. Math. Hungar. 20 (1985), 183–187.

    MATH  Google Scholar 

  16. F. Luca and C. Pomerance, On some problems of Mqkowski-Schinzel and Erdös concerning the arithmetical functions φ and σ, Colloq. Math. 92 (2002), 111–130; Acknowledgement of priority, ibid. 126 (2012), 139.

    Article  MATH  MathSciNet  Google Scholar 

  17. —, On the range of the iterated Euler function, Combinatorial number theory, 101–106, Walter de Gruyter, Berlin, 2009.

    Google Scholar 

  18. H. Maier, On the third iterates of the φ-and σ-function, Colloq. Math. 49 (1984), 123–130.

    MATH  MathSciNet  Google Scholar 

  19. S. S. Pillai, On some functions connected with φ(n), Bull. Amer. Math. Soc. 35 (1929), 832–836.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Pollack, Long gaps between deficient numbers, Acta Arith. 146 (2011), 33–43.

    Article  MATH  MathSciNet  Google Scholar 

  21. -, Two remarks on iterates of Euler’s totient function, Arch. Math. 97 (2011), 449–453.

    Google Scholar 

  22. -, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. 60 (2011), 199–214.

    Google Scholar 

  23. C. Pomerance, On the distribution of amicable numbers, J. Reine Angew Math. 293/294 (1977), 217–222.

    MathSciNet  Google Scholar 

  24. On the composition of the arithmetic functions φ and ϕ, Colloq. Math. 58 (1989), 11–15.

    Google Scholar 

  25. P. Poulet, Nouvelles suites arithmétiques, Sphinx 2 (1932), 53–54.

    Google Scholar 

  26. A. Schinzel, On functions φ(n) and σ(n), Bull. Acad. Polon. Sci. Cl. III, 3 (1955), 415–419.

    MATH  MathSciNet  Google Scholar 

  27. E. Wirsing, Bemerkung zu der Arbeit über volkommene Zahlen, Math. Ann. 137 (1959), 316–318.

    Article  MATH  MathSciNet  Google Scholar 

  28. K. Wooldridge, Values taken many times by Euler’s phi-function, Proc. Amer. Math. Soc. 76 (1979), 229–234.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956, Warsaw, Poland

    Andrzej Schinzel

Authors
  1. Andrzej Schinzel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Eötvös Lóránd University, Pázmány P. sétány 1/c, Budapest, 1117, Hungary

    László Lovász

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Imre Z. Ruzsa  & Vera T. Sós  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2013 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Schinzel, A. (2013). Erdős’s Work on the Sum of Divisors Function and on Euler’s Function. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_21

Download citation

Publish with us

Policies and ethics

Search

Navigation