Transcendence of generating functions whose coefficients are multiplicative

Abstract

Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)={\sum }_{n\ge 1}f(n)z^n\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $\ddot{I}‡(n)$ such that $f(n)=n^k\ddot{I}‡(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, the generating function of a multiplicative function $f:\mathbb{N}\to K$ is either transcendental or rational.

This paper has been withdrawn and replaced with a more current version; see arXiv:1003.2221