An arithmetical function is multiplicative ifis not always zero andwhenever

The functionis completely multiplicative iffor all

Examples:

whereis a fixed, real or complex number, thenis completely multiplicative sinceis called the power function.

The unit functionis completely multiplicative. It can be seen as a special case ofwithor you may notice that

The identity functionis completely multiplicative.

Ifthenand

The Mobius function is multiplicative but not completely multiplicative.

Ifare relatively prime and if eitherorhas a prime square factor then so doesand bothandare zero. If neither has a prime square factor writeand where theandare all distinct primes then andis not completely multiplicative sincebut

The Euler totient functionis multiplicative since ifand thenfor relatively prime. It is not completely multiplicative since

The ordinary productof two arithmetical functionsanddefined byis multiplicative and so iswheneverIfandare completely multiplicative then so areand

For f to be multiplicative we must havesinceWe can cancelsince for someso

is completely multiplicative if