Exerecise 2.34 of Apostol's Analytic number theory is devoted to prove that the Dirichlet inverse of a multiplicative function is multiplicative. The author assumes that $g$ is multiplicative and puts $f=g^{-1}$.
In part(b), author says:
Let $h$ be the uniquely determined multiplicative function which agrees with $f$ at the prime powers.
As I didn't see such a thing in the context of chapter 2, I want to know more about $h$. In fact, how $h$ can be determined and how it is unique? Could someone help me? Thanks!
For a general $f$, the value of $h$ is equal to $f$ at every prime power argument and is determined by multiplicative property at all other arguments - which may there be different to $f$.
$h$ is unique because all the prime power argument $f$ values are required to "seed" the multiplicative process to determine the other values of $h$.