Abstract
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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References
Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)
Buschman, R.G.: lcm-products of number-theoretic functions revisited. Kyungpook Math. J. 39, 159–159 (1999)
Cohen, E.: Representations of even functions (mod \(r\)). II. Cauchy products. Duke Math. J. 26, 165–182 (1959)
Cohen, E.: Arithmetical functions associated with the unitary divisors of an integer. Math. Z. 74, 66–80 (1960)
McCarthy, P.J.: Introduction to Arithmetical Functions. Universitext. Springer, New York (1986)
Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers., 5th edn. Wiley, New York (1991)
Ramanujan, S.: On certain trigonometrical sums and their applications in the theory of numbers. Trans. Camb. Philos. Soc. 22, 259–276 (1918)
Lehmer, D.H.: On a theorem of von Sterneck. Bull. Am. Math. Soc. 37, 723–726 (1931)
Rudin, W.: Functional Analysis. McGraw–Hill, New York (1991)
Tóth, L., Haukkanen, P.: The discrete Fourier transform of \(r\)-even functions. Acta Univ. Sapientiae Math. 3, 5–25 (2011)
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)
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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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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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DOI: https://doi.org/10.1007/s11139-018-0122-8