Advertisement

The Ramanujan Journal

, Volume 47, Issue 3, pp 589–603 | Cite as

Ramanujan expansions of arithmetic functions of several variables

  • László Tóth
Article
  • 148 Downloads

Abstract

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.

Keywords

Ramanujan expansion of arithmetic functions Arithmetic function of several variables Multiplicative function Unitary divisor Unitary Ramanujan sum 

Mathematics Subject Classification

11A25 11N37 

Notes

Acknowledgements

The author thanks the anonymous referee for careful reading of the manuscript and helpful comments.

References

  1. 1.
    Cohen, E.: Arithmetical functions associated with the unitary divisors of an integer. Math. Z. 74, 66–80 (1960)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cohen, E.: Fourier expansions of arithmetical functions. Bull. Am. Math. Soc. 67, 145–147 (1961)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Delange, H.: On Ramanujan expansions of certain arithmetical functions. Acta Arith. 31, 259–270 (1976)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Derbal, A.: La somme des diviseurs unitaires d’un entier dans les progressions arithmétiques (\(\sigma ^*_{k, l}(n)\)). C. R. Math. Acad. Sci. Paris 342, 803–806 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Grytczuk, A.: An identity involving Ramanujan’s sum. Elem. Math. 36, 16–17 (1981)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hölder, O.: Zur Theorie der Kreisteilungsgleichung \(K_m(x)=0\). Prace Mat. -Fiz. 43, 13–23 (1936)zbMATHGoogle Scholar
  7. 7.
    Johnson, K.R.: Unitary analogs of generalized Ramanujan sums. Pac. J. Math. 103, 429–432 (1982)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lucht, L.G.: A survey of Ramanujan expansions. Int. J. Number Theory 6, 1785–1799 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    McCarthy, P.J.: Regular arithmetical convolutions. Portugal Math. 27, 1–13 (1968)MathSciNetzbMATHGoogle Scholar
  10. 10.
    McCarthy, P.J.: Introduction to Arithmetical Functions. Springer, New York (1986)CrossRefGoogle Scholar
  11. 11.
    Postnikov, A.G.: Introduction to analytic number theory. In: Translations of Mathematical Monographs, vol. 68, American Mathematical Society, Providence, RI (1988)Google Scholar
  12. 12.
    Ramanujan, S.: On certain trigonometric sums and their applications in the theory of numbers. Trans. Camb. Philos. Soc. 22, 179–199 (1918)Google Scholar
  13. 13.
    Ram Murty, M.: Ramanujan series for arithmetic functions. Hardy-Ramanujan J. 36, 21–33 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Schwarz, W., Spilker, J.: Arithmetical functions. In: An Introduction to Elementary and Analytic Properties of Arithmetic Functions and to Some of Their Almost-Periodic Properties, London Mathematical Society Lecture Note Series, vol. 184. Cambridge University Press, Cambridge (1994)Google Scholar
  15. 15.
    Sitaramachandrarao, R., Suryanarayana, D.: On \(\sum _{n\le x} \sigma ^*(n)\) and \(\sum _{n\le x} \varphi ^*(n)\). Proc. Am. Math. Soc. 41, 61–66 (1973)CrossRefGoogle Scholar
  16. 16.
    Snellman, J.: The ring of arithmetical functions with unitary convolution: divisorial and topological properties. Arch Math. (Brno) 40, 161–179 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Subbarao, M.V.: A note on the arithmetic functions \(C(n, r)\) and \(C^*(n, r)\). Nieuw Arch. Wisk. 3(14), 237–240 (1966)zbMATHGoogle Scholar
  18. 18.
    Suryanarayana, D.: A property of the unitary analogue of Ramanujan’s sum. Elem. Math. 25, 114 (1970)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Tóth, L.: Remarks on generalized Ramanujan sums and even functions. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 20, 233–238 (2004)Google Scholar
  20. 20.
    Tóth, L.: A survey of the alternating sum-of-divisors function. Acta Univ. Sapientiae. Math. 5, 93–107 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tóth, L.: Multiplicative arithmetic functions of several variables: a survey. In: Rassias, Th.M., Pardalos, P. (eds.) Mathematics Without Boundaries, Surveys in Pure Mathematics. Springer, New York, 2014, 483–514 (2014)Google Scholar
  22. 22.
    Ushiroya, N.: Mean-value theorems for multiplicative arithmetic functions of several variables. Integers 12, 989–1002 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ushiroya, N.: Ramanujan-Fourier series of certain arithmetic functions of two variables. Hardy-Ramanujan J. 39, 1–20 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vaidyanathaswamy, R.: The theory of multiplicative arithmetic functions. Trans. Am. Math. Soc. 33, 579–662 (1931)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wintner, A.: Eratosthenian Averages. Waverly Press, Baltimore (1943)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PécsPécsHungary

Personalised recommendations