Transcendence of generating functions whose coefficients are multiplicative

Details Transcendence of generating functions whose coefficients are multiplicative Transcendence of generating functions whose coefficients are multiplicative

2009-03-30

arxiv.org/abs/0903.5240v3

Mathematics

Number Theory

This paper has been withdrawn and replaced with a more current version; see arXiv:1003.2221

Scientific paper

Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $\chi(n)$ such that $f(n)=n^k \chi(n)$ 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.

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