Unitary Convolution and Generalized MӦbius Function

Main Article Content

Pussadee Yangklan*
Pattira Ruengsinsub
Vichian Laohakosol

Abstract

The concept of unitary convolution in the commutative ring of arithmetic functions, under the two operations of addition and unitary convolution, is investigated. A generalized unitary MӦbius function is defined and its basic properties, which extend those corresponding classical ones are derived. Of particular interest are certain generalized unitary MӦbius inversion formula and some characterizations of multiplicative functions.


Keywords: arithmetic function, Mӧbius inversion formula, multiplicative function, unitary convolution


*Corresponding author:  Tel.: +66 851421272


 E-mail: pussadee_9@hotmail.com

Article Details

Section
Original Research Articles

References

] Sivaramakrishnan, R., 1989. Classical Theory of Arithmetic Functions. Marcel Dekker. New York-Basel.
[2] Cohen, E., 1960. Arithmetical functions associated with the unitary divisors of aninteger. Math. Zeitschr., 74, 66-80.
[3] Cohen, E., 1961. Unitary products of arithmetical functions. Acta Arith., 7, 29-38.
[4] Cohen, E., 1964. Arithmetical notes. X. A class of to tients. Proc. Amer. Math. Soc., 15(4), 534-539.
[5] Horadam, E.M., 1962. Arithmetical function associated with the unitary divisors of a generalized integer. Amer. Math. Monthly, 69, 196-199.
[6] Rao, K.N., 1965. On the unitary an alogues of certain totients. Monatsh. Math., 70, 149-154.
[7] Vaidyanathaswamy, R., 1931. The theory of multiplicative arithmetic functions. Trans. Amer. Math. Soc., 33, 579-662.
[8] Rearick, D., 1968. Operators on algebras of arithmetic functions. Duke. Math. J., 35, 761-766.
[9] Hansen, R.T. and Swanson L.G., 1979. Unitary divisors. Math. Magazine, 52, 217-222.
[10] Johnson, K.R., 1982. Unitary an alog of generalized Ramanujan sums. Pacific J. Math., 103, 429-432.
[11] Hsu, L.C., 1995. A difference-operational approach to the Mӧbius inversion formulas. Fibonacci Quarterly, 33, 169-173.
[12] Hsu, L.C. and Wang J., 1998. Some Mӧbius-type functions and inversions constructed via difference operators. Tamkang J. Math., 29, 89-99.
[13] Schinzel, A., 1998. A property of unitary convolution. Colloq. Math., 78, 93-96.
[14] Buschman, R.G., 2003. Unitary product again, StudiaUniv. Babes-Bolyai, Mathematica, 48, 29-34.