Arithmetic Functions and Geometry

In this expository note, we revisit several classical arithmetic functions - namely Euler's totient function, the divisor sum functions and Dedekind's $\psi$-function - within a unifying algebraic framework that highlights their connections to geometry. This framework builds on prior work involving zeta functions and M\"obius inversion. While our main goal is to provide a clear context for similar constructions in the future, we also make an original observation regarding Dedekind's $\psi$-function.