Transcendence of generating functions whose coefficients are multiplicative

TL;DR

This paper characterizes algebraic generating functions of multiplicative functions over fields of characteristic zero, showing they are either rational or transcendental, with specific structural forms for algebraic cases.

Contribution

It provides a complete classification of algebraic generating functions of multiplicative functions, revealing they are either rational or have a specific multiplicative form.

Findings

Generating functions are either rational or transcendental.

Algebraic generating functions correspond to functions of the form n^k times a periodic multiplicative function.

If not of this form, the generating function is transcendental.

Abstract

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.