Unitary convolution for arithmetical functions in several variables

Alkan, Emre; Zaharescu, Alexandru; Zaki, Mohammad

doi:10.32917/hmj/1147883399

Cited by 6 publications

(4 citation statements)

References 2 publications

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“…For further algebraic properties of the R-algebra A r (R) = {f : N r → R}, where R is an integral domain with respect to the unitary convolution and using the concept of firmly multiplicative functions see E. Alkan, A. Zaharescu, M. Zaki [2].…”

Section: Unitary Convolutionmentioning

confidence: 99%

Multiplicative Arithmetic Functions of Several Variables: A Survey

Tóth¹

2013

Preprint

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We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory and combinatorics. We give a new proof to obtain the asymptotic density of the set of ordered r-tuples of positive integers with pairwise relatively prime components and consider a similar question related to unitary divisors.

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Section: Unitary Convolutionmentioning

confidence: 99%

Multiplicative Arithmetic Functions of Several Variables: A Survey

Tóth¹

2013

Preprint

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“…Yokom [10] studied the prime factorization of arithmetical functions in a certain subring of the regular convolution ring, and determined a discrete valuation subring of the unitary ring of arithmetical functions. Recently, Alkan et al [4] investigated a class of derivations and norms in the ring of arithmetical functions in several variables with unitary convolution. Schwab and Silberberg [8] constructed an extension of (A, +, ·) which is a discrete valuation ring.…”

Section: Introductionmentioning

confidence: 99%

Derivations and generating degrees in the ring of arithmetical functions

Zaharescu

Zaki

2007

Proc Math Sci

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Section: Introductionmentioning

confidence: 99%

“…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]. In this paper we adopt the derivation given in [1] and utilize the method of Rearick [7] to obtain the above mentioned derivation-related property for multiplicative and firmly multiplicative functions, see Theorems 3 and 4.…”

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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Unitary convolution for arithmetical functions in several variables

Cited by 6 publications

References 2 publications

Multiplicative Arithmetic Functions of Several Variables: A Survey

Multiplicative Arithmetic Functions of Several Variables: A Survey

Derivations and generating degrees in the ring of arithmetical functions

Derivation of arithmetical functions under the Dirichlet convolution

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