Distributive property of completely multiplicative functions

Pabhapote, Nittiya; Laohakosol, Vichian

doi:10.1007/s10986-010-9088-y

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…Derivations for arithmetical functions have been presented, e.g., in [1,2,4,3,6]. A certain property of multiplicative type functions in terms of derivations is well known [1,2,4], see also [7,8].…”

Section: Introductionmentioning

confidence: 99%

Derivation of arithmetical functions under the Dirichlet convolution

Haukkanen

2018

Int. J. Number Theory

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We present the group-theoretic structure of the classes of multiplicative and firmly multiplicative arithmetical functions of several variables under the Dirichlet convolution, and we give characterizations of these two classes in terms of a derivation of arithmetical functions. MSC: 11A25f (n 1 ) · · · f (n r ) is firmly multiplicative. On the other hand, the function gcd(n 1 , . . . , n r ) is multiplicative but not firmly multiplicative for r ≥ 2. Further examples can be found, e.g., in [5,10,11]. A survey on multiplicative arithmetical functions of several variables is presented in [11].An arithmetical function f ∈ A r (R) is said to be additive if f (m 1 n 1 , . . . , m r n r ) = f (m 1 , . . . , m r ) + f (n 1 , . . . , n r ) for all positive integers m 1 , . . . , m r and n 1 , . . . , n r with (m 1 · · · m r , n 1 · · · n r ) = 1, and an arithmetical function f ∈ A r (R) is said to be completely additive if f (m 1 n 1 , . . . , m r n r ) = f (m 1 , . . . , m r ) + f (n 1 , . . . , n r )

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Section: Introductionmentioning

confidence: 99%

Derivation of arithmetical functions under the Dirichlet convolution

Haukkanen

2018

Int. J. Number Theory

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“…In this work we will give yet another answer to the comment transcribed verbatim above from the perspective of number theory. Using the SCSs [1] and Gödel relabeling of quantum states [18] we obtain a quantum mechanical representation of a generalization of the Möbius function: the Fleck function [22][23][24][25][26], sometimes referred to as the Souriau-Hsu-Möbius function (SHM) or generalized Fleck function (GF) for non-negative integer values (a generalization of the Popovici function) [27][28][29]. Throughout this work we will refer to it simply as F = Fleck function.…”

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confidence: 99%

Spin coherent states, binomial convolution and a generalization of the Möbius function

Neto¹

2012

J. Phys. A: Math. Theor.

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By making use of a Gödel-type relabeling of quantum states we show that spin coherent states play a fundamental role in number theory. We generalize the representation of the Möbius function obtained in Spector (1990 Commun. Math. Phys. 127 239) by giving a quantum mechanical interpretation of a generalization of the Möbius function: the Fleck function. We also show that inversion convolution theorem for the Liouville function and some key relations giving the Möbius inversion theorem can be understood from the orthogonality properties of the spin coherent states. Our results show a fruitful interplay of quantum mechanics and number theory.

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Some properties of totients

Haukkanen

2024

Ramanujan J

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A arithmetical function f is said to be a totient if there exist completely multiplicative functions $$f_t$$ f t and $$f_v$$ f v such that$$ f=f_t*f_v^{-1}, $$ f = f t ∗ f v - 1 , where $$*$$ ∗ is the Dirichlet convolution. Euler’s $$\phi $$ ϕ -function is an important example of a totient. In this paper we find the structure of the usual product of two totients, the usual integer power of totients, the usual product of a totient and a specially multiplicative function and the usual product of a totient and a completely multiplicative function. These results are derived with the aid of generating series. We also provide some distributive-like characterizations of totients involving the usual product and the Dirichlet convolution of arithmetical functions. They give as corollaries characterizations of completely multiplicative functions.

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Distributive property of completely multiplicative functions

Cited by 3 publications

References 9 publications

Derivation of arithmetical functions under the Dirichlet convolution

Derivation of arithmetical functions under the Dirichlet convolution

Spin coherent states, binomial convolution and a generalization of the Möbius function

Some properties of totients

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