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The Ramanujan Journal

, Volume 27, Issue 1, pp 71–88 | Cite as

An analogue of Ramanujan’s sum with respect to regular integers (modr)

  • Pentti HaukkanenEmail author
  • László Tóth
Article
  • 124 Downloads

Abstract

An integer a is said to be regular (modr) if there exists an integer x such that a 2 xa (mod r). In this paper we introduce an analogue of Ramanujan’s sum with respect to regular integers (modr) and show that this analogue possesses properties similar to those of the usual Ramanujan’s sum.

Keywords

Ramanujan’s sum Regular integer Arithmetical convolution Even function Discrete Fourier transform Multiplicative function Mean value Dirichlet series 

Mathematics Subject Classification (2000)

11A25 11L03 

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References

  1. 1.
    Alkam, O., Osba, E.A.: On the regular elements in Z n. Turk. J. Math. 32, 31–39 (2008) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Anderson, D.R., Apostol, T.M.: The evaluation of Ramanujan’s sum and generalizations. Duke Math. J. 20, 211–216 (1953) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Apostol, T.M.: Arithmetical properties of generalized Ramanujan sums. Pac. J. Math. 41, 281–293 (1972) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, Berlin (1976) zbMATHGoogle Scholar
  5. 5.
    Bundschuh, P., Hsu, L.C., Shiue, P.J.-S.: Generalized Möbius inversion-theoretical and computational aspects. Fibonacci Q. 44, 109–116 (2006) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Cohen, E.: A class of arithmetical functions. Proc. Natl. Acad. Sci. USA 41, 939–944 (1955) CrossRefzbMATHGoogle Scholar
  7. 7.
    Cohen, E.: Arithmetical functions associated with the unitary divisors of an integer. Math. Z. 74, 66–80 (1960) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cohen, E.: Unitary products of arithmetic functions. Acta Arith. 7, 29–38 (1961) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Haukkanen, P.: Classical arithmetical identities involving a generalization of Ramanujan’s sum. Ann. Acad. Sci. Fenn., Ser. A 1 Math. Diss. 68, 1–69 (1988) MathSciNetGoogle Scholar
  10. 10.
    Haukkanen, P.: On some set-reduced arithmetical sums. Indian J. Math. 39, 147–158 (1997) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Haukkanen, P.: An elementary linear algebraic approach to even functions (mod r). Nieuw Arch. Wiskd., (5) 2, 29–31 (2001) MathSciNetGoogle Scholar
  12. 12.
    Haukkanen, P.: On an inequality related to the Legendre totient function. JIPAM. J. Inequal. Pure Appl. Math. 3, 37 (2002), 6 pp. MathSciNetGoogle Scholar
  13. 13.
    Haukkanen, P.: Discrete Ramanujan–Fourier transform of even functions (modr). Indian J. Math. Math. Sci. 3, 75–80 (2007) MathSciNetGoogle Scholar
  14. 14.
    He, T.-X., Hsu, L.C., Shiue, P.J.-S.: On generalised Möbius inversion formulas. Bull. Aust. Math. Soc. 73, 79–88 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    McCarthy, P.J.: Introduction to Arithmetical Functions. Universitext. Springer, Berlin (1986) CrossRefzbMATHGoogle Scholar
  16. 16.
    Mednykh, A.D., Nedela, R.: Enumeration of unrooted maps with given genus. J. Comb. Theory, Ser. B 96, 706–729 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Montgomery, H.L., Vaughan, R.C., Multiplicative Number Theory, I.: Classical Theory. Cambridge Studies in Advanced Mathematics, vol. 97. Cambridge University Press, Cambridge (2007) zbMATHGoogle Scholar
  18. 18.
    Ramanujan, S.: On certain trigonometrical sums and their applications in the theory of numbers. Trans. Camb. Philos. Soc. 22, 259–276 (1918) Google Scholar
  19. 19.
    Rearick, D.: Semi-multiplicative functions. Duke Math. J. 33, 49–53 (1966) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Rearick, D.: Correlation of semi-multiplicative functions. Duke Math. J. 33, 623–627 (1966) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Samadi, S., Ahmad, M.O., Swamy, M.N.S.: Ramanujan sums and discrete Fourier transforms. IEEE Signal Process. Lett. 12, 293–296 (2005) CrossRefGoogle Scholar
  22. 22.
    Sándor, J., Crstici, B.: Handbook of Number Theory II. Kluwer Academic, Dordrecht (2004) CrossRefzbMATHGoogle Scholar
  23. 23.
    Schramm, W.: The Fourier transform of functions of the greatest common divisor. Integers 8, #A50 (2008), 7 pp. MathSciNetGoogle Scholar
  24. 24.
    Schwarz, W., Spilker, J.: Arithmetical Functions. London Mathematical Society Lecture Note Series, vol. 184. Cambridge University Press, Cambridge (1994) zbMATHGoogle Scholar
  25. 25.
    Selberg, A.: Remarks on multiplicative functions. In: Number Theory Day. Proc. Conf., Rockefeller Univ., New York, 1976, pp. 232–241. Springer, Berlin (1977) CrossRefGoogle Scholar
  26. 26.
    Sivaramakrishnan, R.: Classical Theory of Arithmetic Functions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 126. Marcel Dekker, New York (1986) Google Scholar
  27. 27.
    Spilker, J.: Eine einheitliche Methode zur Behandlung einer linearen Kongruenz mit Nebenbedingungen. Elem. Math. 51, 107–116 (1996) MathSciNetGoogle Scholar
  28. 28.
    Subbarao, M.V., Harris, V.C.: A new generalization of Ramanujan’s sum. J. Lond. Math. Soc. 41, 595–604 (1966) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Tóth, L.: Remarks on generalized Ramanujan sums and even functions. Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 20, 233–238 (2004) zbMATHMathSciNetGoogle Scholar
  30. 30.
    Tóth, L.: Regular integers (mod n). Annales Univ. Sci. Budapest., Sect. Comp. 29, 263–275 (2008) zbMATHGoogle Scholar
  31. 31.
    Tóth, L.: A gcd-sum function over regular integers modulo n. J. Integer Seq. 12, 09.2.5 (2009) Google Scholar
  32. 32.
    Yamasaki, Y.: Arithmetical properties of multiple Ramanujan sums. Ramanujan J. 21, 241–261 (2010) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TampereTampereFinland
  2. 2.Institute of Mathematics and InformaticsUniversity of PécsPécsHungary

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