Transcendence of generating functions whose coefficients are multiplicative

In this paper, we give a new proof and an extension of the following result of B'ezivin. Let $f:\text{B}N\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$foritsgeneratingseries\mathrm{.}Supposethat$F(z)$isalgebraicover$K(z)$\mathrm{.}Theneitherthereisanaturalnumber$k$ 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 $K=\text{B}C$, 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.