Arithmetic Functions and Geometry
Andrew Kobin
TL;DR
The paper develops an algebraic and geometric lens on classical arithmetic functions by embedding them in incidence algebras and motivic frameworks. It extends the totient, divisor-sum, and psi-functions to number fields, varieties, and arithmetic schemes, deriving multiplicativity properties and Dirichlet-series identities that mirror their classical counterparts, e.g. $\sum \phi(n)/n^s=\zeta(s-1)/\zeta(s)$ and $\sum \psi(n)/n^s=\zeta(s)\zeta(s-1)/\zeta(2s)$. A key theme is that these functions arise as shadows of geometric objects such as motivic zeta functions, configuration spaces, and Grothendieck rings, yielding a unifying perspective and motivating future geometric constructions of arithmetic invariants. The work also presents an original viewpoint on Dedekind’s $\psi$-function through Witt-vector group counts and discusses global, motivic, and scheme-theoretic generalizations, highlighting the potential of arithmetic geometry to illuminate classical number-theoretic identities and their extensions. Overall, it offers a coherent, non-original (with one exception) synthesis that links arithmetic functions to geometry and motivates further exploration of arithmetic invariants in a geometric setting.
Abstract
In this expository note, we revisit several classical arithmetic functions - namely Euler's totient function, the divisor sum functions and Dedekind's $ψ$-function - within a unifying algebraic framework that highlights their connections to geometry. This framework builds on prior work involving zeta functions and Möbius 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 $ψ$-function.