Transcendence of generating functions whose coefficients are multiplicative
Jason P. Bell, Nils Bruin, Michael James Coons · Transactions of the American Mathematical Society · 2011
In this paper, we give a new proof and an extension of the following result of Bézivin. Let $f:\mathbb {N}\to K$ 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. If $F(z)$ is algebraic, 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, the generating series of a multiplicative function taking values in a field of characteristic zero is either transcendental or rational. For $K=\mathbb {C}$, we also prove that if the generating series of a multiplicative function is $D$–finite, then it must either be transcendental or rational.