Transcendence of Power Series for Some Number Theoretic Functions

TL;DR

This paper provides a new proof of Fatou's theorem and applies it to demonstrate the transcendence of certain power series associated with multiplicative functions like Liouville's and Möbius functions.

Contribution

It introduces a novel proof of Fatou's theorem and establishes the transcendence of power series for specific multiplicative functions, extending previous results in number theory.

Findings

New proof of Fatou's theorem for algebraic functions with bounded integer coefficients.

Proves the transcendence of power series for non-trivial multiplicative functions such as Liouville's and Möbius functions.

Shows that these series are transcendental over the polynomial ring z[z].

Abstract

We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely multiplicative function from $\mathbb{N}$ to $\{-1,1\}$, the series $\sum_{n=1}^\infty f(n)z^n$ is transcendental over $\mathbb{Z}[z]$; in particular, $\sum_{n=1}^\infty \lambda(n)z^n$ is transcendental, where $\lambda$ is Liouville's function. The transcendence of $\sum_{n=1}^\infty \mu(n)z^n$ is also proved.