I'm currently reading Apostol's analytic number theory, Chapter 2 on multiplicative functions. While the current exposition is nice, I can't help but feel that there has to been some algebraic structure lurking behind how nice the algebra of $\mu$, $\phi$, and the Dirichlet convolution $\star$ operators are.
Is there some place I can find an elementary, but rigorous introduction to these developed entirely algebraically (I feel that the algebraic structure ought to be lattice-y or ring-ey), which is then "instantiated" on number theory? Would I find this developed in, say, an algebraic number theory text?
Apostol presents the result that the arithmetic functions with respect to addition and Dirichlet product form a factorial ring. This is shown in many texts of number theory, but it does not use particular methods from algebraic number theory.