An arithmetical function is multiplicative if
is not always zero and
whenever 
The functionis completely multiplicative iffor all
Examples:
where
is a fixed, real or complex number, then
is completely multiplicative since
is called the power function.
The unit function
is completely multiplicative. It can be seen as a special case ofwith
or you may notice that
The identity function
is completely multiplicative.
If
then
and
The Mobius function is multiplicative but not completely multiplicative.
If
are relatively prime and if either
or
has a prime square factor then so does
and both
and
are zero. If neither has a prime square factor write
and 
where the
and
are all distinct primes then
and
is not completely multiplicative since
but 
The Euler totient function
is multiplicative since if
and
then
for relatively prime. It is not completely multiplicative since
The ordinary product
of two arithmetical functionsand
defined by
is multiplicative and so is
whenever
Ifandare completely multiplicative then so areand
For f to be multiplicative we must have
since
We can cancel
since for some
so
is completely multiplicative if