A Characterization of Completely Multiplicative Arithmetic Functions

Characterizing completely multiplicative functions bygeneralized Möbius functions

“…Proof. From hypothesis, we have Now proceed by induction noting, as in the lemma of Carroll [4], that f (1) = 1 and f (p i ) = f (p) i (i = 1,...,k− 1) imply f −1 (p i ) = 0 (i = 2, 3,...,k− 1). We thus have…”

A Representation of Multiplicative Arithmetic Functions by Symmetric Polynomials

“…Another question arises from the fact that the UFD A * has a rich ideal structure [2,32]. Is there a representation of these ideals in terms of symmetric polynomials?…”

“…In an earlier work which appeared in [23] (also see [2] and [3] ), a construction was given which provided a q-th root, q ∈ Q, for every element of the WIP-module, thus giving an effective construction of the divisible closure (injective hull) of the WIP-module. So, in particular, providing a construction for the q−roots of any multiplicative function.…”

“…In Section 9, we state the divisible embedding theorem for the WIP-module [23], [2] and apply it to the MF's; that is, we give a formula for producing q − th-roots, q ∈ Q, with respect to the convolution product for each Schur-hook polynomial in the WIP-module. This construction translates to the convolution group of multiplicative functions, embedding it in its divisible closure.…”