Arithmetical Functions in Several Variables

Abstract

In this paper we investigate the ring Ar(R) of arithmetical functions in r variables over an integral domain R. We study a class of absolute values, and a class of derivations on Ar(R). We show that a certain extension of Ar(R) is a discrete valuation ring. We also investigate the metric structure of the ring Ar(R).

An algebraic independence result for Euler products of finite degree

“…which is a discrete valuation ring. Alkan and the authors [1] generalized this construction and provided a family of extensions of A r which are discrete valuation rings. For other work on rings of arithmetical functions the reader is referred to [5], [6], [9], [12], [13], [10], [11], [2].…”

“…For other work on rings of arithmetical functions the reader is referred to [5], [6], [9], [12], [13], [10], [11], [2]. In [1], it was shown that for any completely additive arithmetical function ψ ∈ A r , the map D ψ : A r → A r defined by D ψ (f )(n 1 , . .…”

“…, n r ∈ N, is a derivation on A r . It was also proved in [1] that for any multiplicative function f ∈ A r , any completely additive function ψ ∈ A r , and any n 1 , . .…”

Derivations and generating degrees in the ring of arithmetical functions

“…Further results, in a more abstract categorical setting, have been obtained by Schwab in [6] and [7]. In [3], a class of absolute values and a family of derivations on the ring of arithmetical functions in several variables, with the analogue of Dirichlet convolution as multiplication, are defined and studied. Let R be an integral domain, r a positive integer, and A r (R) = {f : N r → R}.…”

“…The topologies obtained from the valuation constructed in [8] and its generalizations defined in [3] play an important role in our present investigation. We consider a natural family of subrings B r,k,p (R) of A r (R), which are closed in each of these topologies.…”