Abstract
AbstractA 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.
Publisher
Springer Science and Business Media LLC
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