Ah, this is a nice example why it is not a good idea to read in the middle of the a book without knowing what's been treated before: they're talking there (page 37 in Apostol's book ) about inverses wrt to Dirichlet's Multiplication, NOT usual inverses of bijective functions (which should be obvious also because the Euler's Totient Functions is far from being bijective).

Begin at page 29 reading about Dirichlet's product and its main properties. It's very beautiful stuff.

Tonio