BASIC PROPERTY OF GENERALIZEDCOMPLETELY MULTIPLICATIVEFUNCTIONS

Abstract

An arithmetic function
f
is said to be completely multiplicative if
f
(1) = 1 and
f
(
mn
) =
f
(
m
)
f
(
n
) for all positive integers
m
and
n
.
In this paper, we define that an arithmetic function
f
is a generalized
completely multiplicative function if
f
(1) = 1 and there is a completely
multiplicative function
f
b
such that
f
(
mn
) =
f
(
m
)
f
b
(
n
)
f
(
n
)
f
b
(
m
)
for all
positive integers
m
and
n
. We consider some basic structure properties of
these functions. The functions
v
(
n
) =
n
n
and Exp
D
are examples of gen-
eralized completely multiplicative functions, where
D
is the arithmetic
derivative