Properties
A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a product of powers of distinct primes, say n = pa qb ..., then f(n) = f(p)a f(q)b ...
While the Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative.
There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function f multiplicative then is completely multiplicative if and only if the Dirichlet inverse is where is the Mobius function.
Completely multiplicative functions also satisfy a pseudo-associative law. If f is completely multiplicative then
where * represents the Dirichlet product and represents pointwise multiplication. One consequence of this is that for any completely multiplicative function f one has
Here is the divisor function.
Read more about this topic: Completely Multiplicative Function
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