Extended arithmetic functions

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In this paper, we give an attempt to extend some arithmetic properties such as multiplicativity and convolution products to the setting of operator theory and we provide significant examples which are of interest in number theory. We also give a representation of the Euler differential operator by means of the Euler totient arithmetic function and idempotent elements of some associative unital algebra.

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  1. 1.

    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)

    Google Scholar 

  2. 2.

    Buschman, R.G.: lcm-products of number-theoretic functions revisited. Kyungpook Math. J. 39, 159–159 (1999)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Cohen, E.: Representations of even functions (mod \(r\)). II. Cauchy products. Duke Math. J. 26, 165–182 (1959)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cohen, E.: Arithmetical functions associated with the unitary divisors of an integer. Math. Z. 74, 66–80 (1960)

    MathSciNet  Article  Google Scholar 

  5. 5.

    McCarthy, P.J.: Introduction to Arithmetical Functions. Universitext. Springer, New York (1986)

    Google Scholar 

  6. 6.

    Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers., 5th edn. Wiley, New York (1991)

    Google Scholar 

  7. 7.

    Ramanujan, S.: On certain trigonometrical sums and their applications in the theory of numbers. Trans. Camb. Philos. Soc. 22, 259–276 (1918)

    Google Scholar 

  8. 8.

    Lehmer, D.H.: On a theorem of von Sterneck. Bull. Am. Math. Soc. 37, 723–726 (1931)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Rudin, W.: Functional Analysis. McGraw–Hill, New York (1991)

    Google Scholar 

  10. 10.

    Tóth, L., Haukkanen, P.: The discrete Fourier transform of \(r\)-even functions. Acta Univ. Sapientiae Math. 3, 5–25 (2011)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Tóth, L.: Multiplicative arithmetic functions of several variables: a survey. In: Rassias, T., Pardalos, P. (eds.) Mathematics Without Boundaries, pp. 483–514. Springer, New York (2014)

    Google Scholar 

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The authors would like to thank the referee for his valuable comments which helped to improve the paper.

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Correspondence to Fethi Bouzeffour.

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Fethi Bouzeffour and Mubariz Garayev would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this research through the Research Group No. RGP-VPP-323.

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Bouzeffour, F., Jedidi, W. & Garayev, M. Extended arithmetic functions. Ramanujan J 51, 593–609 (2020). https://doi.org/10.1007/s11139-018-0122-8

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  • Arithmetic functions
  • Convolution product
  • Idempotent

Mathematics Subject Classification

  • 11A25
  • 16U99