Applications of differential algebra to algebraic independence of arithmetic functions

TL;DR

This paper applies differential algebra techniques to unify and extend results on the algebraic independence of arithmetic functions and Dirichlet series, revealing new counterexamples to existing claims.

Contribution

It generalizes proofs of algebraic independence results using Ax's differential Schanuel conjecture and identifies counterexamples to previous literature.

Findings

Unified proofs of algebraic independence results

Counterexamples to some existing results

Extension of differential algebra methods to arithmetic functions

Abstract

We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.