# multiplicative function

A function f(*n*) defined on the positive integers
is multiplicative if f(*nm*)=f(*n*)f(*m*)
whenever *n* and *m* are relatively prime.
Clearly f(1) must be 0 or 1. If f(1)=0, then
f(*n*)=0 for all positive integers *n*.
So some authors require that f(1) be non-zero.

If f(*n*) is multiplicative and we factor
*n* into distinct primes as
*n*=*p*_{1}^{a1}^{.}
*p*_{2}^{a2}^{.}
...^{.}*p*_{k}^{ak},
then

f(n) = f(p_{1}^{a1})^{.}f(p_{2}^{a2})^{.}...^{.}f(p_{k}^{ak}).

Finally, if f(*n*) is multiplicative, then so is
the function F(*n*) = sum of f(*i*) (where the sum
is taken over the divisors *i* of *n*).

**See Also:** CompletelyMultiplicative, EulersPhi

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