Multiplicative function nature and common multiplicative function

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Multiplicative function of the nature of

We have already mentioned, the multiplicative function, i.e. the condition \ (\ forall a, b \ in Z _ +, gcd (a, b) = 1 \ Leftrightarrow \ boldsymbol f (a) \ boldsymbol f (b) = \ boldsymbol f (ab) \ ) function \ (\ boldsymbol f \)

In particular, for \ (\ forall a, b \ in Z _ +, \ boldsymbol f (a) \ boldsymbol f (b) = \ boldsymbol f (ab) \) function \ (\ boldsymbol f \) We called complete multiplicative function

Of course, there are multiplicative function on the corresponding must have additive function: for \ (\ forall a, b \ in Z _ +, gcd (a, b) = 1 \ Leftrightarrow f (a) + f (b) = f ( a + b) \)

Corresponding to, if not required to meet \ (gcd (a, b) = 1 \) called full additive function

Obviously, for the constant \ (C (n) \) additive function \ (\ Alpha (n-) \) , apparently \ (C ^ \ alpha (n ) \) function is a multiplicative function; if \ (\ Alpha (n-) \) completely additive function, the \ (C ^ \ alpha (n ) \) function complete multiplicative function