Applications of differential algebra to algebraic independence of 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.

(v.2) General improvements on the presentation. Corrected some mistakes in the proof of Theorem~5.5 in the first version. (v.3) Added an addendum showing that the assumption on continuity of derivations (with respect to the norm topology) of our results can be dropped