A study on some generalized multiplicative and generalized additive arithmetic functions

D. Bhattacharjee
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 32—44
DOI: 10.7546/nntdm.2021.27.1.32-44
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Authors and affiliations

D. Bhattacharjee
Department of Mathematics, North-Eastern Hill University
Permanent Campus, Shillong-793022, India


In this paper by an arithmetic function we shall mean a real-valued function on the set of positive integers. We recall the definitions of some common arithmetic convolutions. We also recall the definitions of a multiplicative function, a generalized multiplicative function (or briefly a GM-function), an additive function and a generalized additive function (or briefly a GA-function). We shall study in details some properties of GM-functions as well as GA-functions using some particular arithmetic convolutions namely the Narkiewicz’s A-product and the author’s B-product. We conclude our discussion with some examples.


  • Arithmetic function
  • Multiplicative function
  • Arithmetic convolution
  • Dirichlet convolution
  • Narkiewicz’s A-product
  • B-product
  • Multiplicative B-product
  • GA-function
  • GM-function

2010 Mathematics Subject Classification

  • 11A25


  1. Apostol, T. M. (1976). Introduction to Analytic Number Theory, Springer-Verlag, New York.
  2. Bell, E. T. (1915). Arithmetical Theory of Certain Numerical Functions. Volume 1,
    University of Washington.
  3. Bhattacharjee, D. (1997). B-product and its properties. Bulletin of Pure and Applied Sciences, 16(1), 77–83.
  4. Bhattacharjee, D. (1998). Multiplicative B-product and its properties. Georgian
    Mathematical Journal, 5(4), 315–320.
  5. Bhattacharjee, D. (2002). A Generalized B-product and its properties, Bulletin of the Allahabad Mathematical Society, 17, 17–21.
  6. Bhattacharjee, D. (2005). Multiplicative 𝐾𝐵-product and its properties. Bulletin of the Calcutta Mathematical Society, 97(2), 153–162.
  7. Bhattacharjee, D. (2007). On Some New Arithmetical Convolutions. Proceedings of the 2007 International Conference on High Performance Computing Networking and Communication Systems (HPCNCS-07), 9–12 July 2007, Orlando, FL, USA, 26–30.
  8. Bhattacharjee, D., & Saikia, P. K. (2012). On Some New Class of Arithmetic Convolutions Involving Arbitrary Sets of Integers. Romanian Journal of Mathematics and Computer Science, 2(2), 23–35.
  9. Chawla, L. M. (1973). Distributive arithmetical functions of two or more variables. Journal of Natural Sciences Math., 13, 263–270.
  10. Pellegrino, F. (1953) Sviluppi moderni del Calcolo numerico integrale di Michele Cipolla. IV Congresso dell’ Unione Matematica Italiana-Taormina, 2, 161–168.
  11. Cohen, E. (1959).Arithmetical functions associated with arbitrary sets of integers, Acta Arithmetica, 5, 407–415.
  12. Davison, T. M. K. (1966). On Arithmetic Convolutions. Canadian Mathematical Bulletin, 9, 287–296.
  13. Dickson, L. E.(1919).History of the Theory of Numbers, Washington: Carnegie Institution.
  14. Gioia, A. A. (1965).The K-product of the arithmetic functions, Canadian Journal of Mathematics, 17, 970–976.
  15. Haukkanen, P. (1992). A note on generalized multiplicative and generalized additive arithmetic functions. The Mathematics Student, 61(1–4), 113–116.
  16. Haukkanen, P. (1996). On a Binomial Convolution of Arithmetic Functions. Nieuw Archief voor Wiskunde, 14(2), 209–216.
  17. Lehmer, D. H. (1931). A New Calculus of Numerical Functions. American Journal of Mathematics, 53, 843–854.
  18. McCarthy, P. J. (1986). Introduction to Arithmetical Functions, Springer-Verlag, New York.
  19. Narkiewicz, W. (1963). On a class of arithmetical convolutions. Colloquium
    Mathematicum, 10, 81–94.
  20. Scheid, H. (1969). Einige Ringe zahlentheoretischer Funktionen. Journal für die reine und angewandte Mathematik, 1969(237), 1–11.
  21. Sivaramakrishnan, R. (1989). Classical Theory of Arithmetical Functions, Monographs and Textbooks in Pure and Applied Mathematics, 126, Marcel-Dekker Inc., New York.
  22. Subbarao, M. V. (1972). On Some Arithmetical Convolutions, Lecture Notes in
    Mathematics, Springer, 251, 247–271.
  23. Tóth, L. (2002). On a Class of Arithmetic Convolutions involving Arbitrary Sets of
    Integers. Mathematica Pannonica, 13, 249–263.
  24. Tóth, L. (2004). On a Certain Arithmetic Functions involving Exponential Divisors. Annales Univ. Sci. Budapest., Sect. Comp., 24, 285–294.
  25. Vaidyanathaswamy, R. (1931). The Theory of Multiplicative Arithmetical Functions. Transactions of the American Mathematical Society, 33, 579–662.
  26. Zafrullah, M. (1988). On generalized multiplicative functions. Journal of Natural Sciences and Mathematics, 28, 257–268.

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Bhattacharjee, D. (2021). A study on some generalized multiplicative and generalized additive arithmetic functions. Notes on Number Theory and Discrete Mathematics, 27(1), 32-44, doi: 10.7546/nntdm.2021.27.1.32-44.

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