Transcendence of generating functions whose coefficients are multiplicative

Abstract

In this paper, we give a new proof and an extension of the following result of B\'ezivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for its generating series. Suppose that $F(z)$ is algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $\chi(n)$ such that $f(n)=n^k \chi(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, $F(z)$ is either transcendental or rational. For $K=\B{C}$, we also prove that if $F(z)$ is a $D$-finite generating series of a multiplicative function, then $F(z)$ is either transcendental or rational.