Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 648165, 15 pages
http://dx.doi.org/10.1155/2010/648165
Research Article

Combinatorial Aspects of the Generalized Euler's Totient

1Department of Mathematics, University of the Thai Chamber of Commerce, Bangkok 10400, Thailand
2Department of Mathematics, Faculty of Science and Institute for Advanced Studies, Kasetsart University, Bangkok 10900, Thailand

Received 24 April 2010; Revised 29 July 2010; Accepted 6 August 2010

Academic Editor: Pentti Haukkanen

Copyright © 2010 Nittiya Pabhapote and Vichian Laohakosol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.
  2. R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, vol. 126 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1989.
  3. J.-M. Souriau, “Généralisation de certaines formules arithmétiques d'inversion. Applications,” Revue Sci, vol. 82, pp. 204–211, 1944. View at Google Scholar · View at Zentralblatt MATH
  4. T. C. Brown, L. C. Hsu, J. Wang, and P. J.-S. Shiue, “On a certain kind of generalized number-theoretical Möbius function,” The Mathematical Scientist, vol. 25, no. 2, pp. 72–77, 2000. View at Google Scholar · View at Zentralblatt MATH
  5. J. Sándor and B. Crstici, Handbook of Number Theory. II, Kluwer Academic, Dordrecht, The Netherlands, 2004.
  6. J. Wang and L. C. Hsu, “On certain generalized Euler-type totients and Möbius-type functions,” Dalian University of Technology, China, preprint.
  7. P. Haukkanen, “A further combinatorial number-theoretic extension of Euler's totient,” Journal of Mathematical Research and Exposition, vol. 17, no. 4, pp. 519–523, 1997. View at Google Scholar · View at Zentralblatt MATH
  8. V. Laohakosol and N. Pabhapote, “Properties of rational arithmetic functions,” International Journal of Mathematics and Mathematical Sciences, no. 24, pp. 3997–4017, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. V. Laohakosol, P. Ruengsinsub, and N. Pabhapote, “Ramanujan sums via generalized Möbius functions and applications,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 60528, 35 pages, 2006. View at Google Scholar · View at Zentralblatt MATH
  10. H. N. Shapiro, Introduction to the Theory of Numbers, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1983.
  11. D. Rearick, “Operators on algebras of arithmetic functions,” Duke Mathematical Journal, vol. 35, pp. 761–766, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. D. Rearick, “The trigonometry of numbers,” Duke Mathematical Journal, vol. 35, pp. 767–776, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. V. Laohakosol, N. Pabhapote, and W. Wechwiriyakul, “Logarithmic operators and characterizations of completely multiplicative functions,” Southeast Asian Bulletin of Mathematics, vol. 25, no. 2, pp. 273–281, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. P. Haukkanen, “On the real powers of completely multiplicative arithmetical functions,” Nieuw Archief voor Wiskunde, vol. 15, no. 1-2, pp. 73–77, 1997. View at Google Scholar · View at Zentralblatt MATH
  15. V. L. Klee, Jr., “A generalization of Euler's ϕ-function,” The American Mathematical Monthly, vol. 55, pp. 358–359, 1948. View at Google Scholar
  16. P. G. Garcia and Steve Ligh, “A generalization of Euler's ϕ function,” The Fibonacci Quarterly, vol. 21, no. 1, pp. 26–28, 1983. View at Google Scholar
  17. L. Tóth, “A note a generalization of Euler's ϕ function,” The Fibonacci Quarterly, vol. 25, pp. 241–244, 1982. View at Google Scholar
  18. D. L. Goldsmith, “A remark about Euler's function,” The American Mathematical Monthly, vol. 76, no. 2, pp. 182–184, 1969. View at Publisher · View at Google Scholar
  19. W. J. LeVeque, Topics in Number Theory. Vols. 1 and 2, Addison-Wesley, Reading, Mass, USA, 1956.
  20. D. Suryanarayana, “New inversion properties of μ and μ,” Elemente der Mathematik, vol. 26, pp. 136–138, 1971. View at Google Scholar · View at Zentralblatt MATH
  21. P. J. McCarthy, “On a certain family of arithmetic functions,” The American Mathematical Monthly, vol. 65, pp. 586–590, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. L. Carlitz, “Note on an arithmetic function,” The American Mathematical Monthly, vol. 59, pp. 386–387, 1952. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. L. E. Dickson, “A generalization of Fermat's Theorem,” in Collected Mathematical Papers, vol. 2, Chelsea Publishing, New York, NY, USA, 1975. View at Google Scholar
  24. L. E. Dickson, “Generalizations of Fermat's Theorem,” in History of the Theory of Numbers, vol. 1, Chelsea Publishing, New York, NY, USA, 1999. View at Google Scholar
  25. P. J. McCarthy, “On an arithmetic function,” Monatshefte für Mathematik, vol. 63, pp. 228–230, 1959. View at Google Scholar · View at Zentralblatt MATH
  26. R. Lidl and H. Niederreiter, Finite Fields, vol. 20 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass, USA, 1983.