A multiplicative number theoretic function is a number theoretic function 
that has the property
 |
(1)
|
for all pairs of relatively prime positive integers 
and 
.
If
 |
(2)
|
is the prime factorization of a number 
,
then
 |
(3)
|
Multiplicative number theoretic functions satisfy the amazing identity
 |  | ![product_(p)[sum_(k=0)^(infty)f(p^k)p^(-ks)]](/images/equations/MultiplicativeNumberTheoreticFunction/Inline7.svg) |
(4)
|
 |  | ![product_(p)[1+f(p)p^(-s)+f(p^2)p^(-2s)+...],](/images/equations/MultiplicativeNumberTheoreticFunction/Inline10.svg) |
(5)
|
where the product is over the primes.
