Transcendence of generating functions whose coefficients are multiplicative

TL;DR

This paper extends Bézivin's result by proving that algebraic generating functions of multiplicative functions are either rational or transcendental, and explores the conditions under which these functions are $D$-finite.

Contribution

The paper provides a new proof and extension of Bézivin's theorem, characterizing algebraic and $D$-finite generating functions of multiplicative functions.

Findings

Algebraic generating functions of multiplicative functions are either rational or transcendental.

If the generating function is $D$-finite, it is either rational or transcendental.

Multiplicative functions with algebraic generating functions are either of the form $n^k imes$ periodic function or eventually zero.

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.