Derivation of arithmetical functions under the Dirichlet convolution

Abstract

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 )

Formal power series in several variables

“…The theory of multiplicative arithmetical functions of several variables originates to the seminal paper of Vaidyanathaswamy [12] from 1931. Recently, this theory has been developed, e.g., in [3,6,11]. Formal power series is a useful tool in this theory as was already noted in [12].…”

Quotients of arithmetical functions under the Dirichlet convolution

“…Remark 3.1. In (7), the summation over k is finite. It suffices that k runs through the integers from 1 to Ω(n) + 1, where Ω(n) is the total number of prime factors of n, each being counted according to its multiplicity, with Ω(1) = 0.…”

Untitled

“…Remark 3.1. In (7), the summation over k is finite. It suffices that k runs through the integers from 1 to Ω(n) + 1, where Ω(n) is the total number of prime factors of n, each being counted according to its multiplicity, with Ω(1) = 0.…”

“…Proof. We proceed by induction on n. For n = 1, the right side of ( 7) is g(c)/h(b) and hence (7) holds.…”