Transcendence of Power Series for Some Number Theoretic Functions

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, ${\sum }_{n=1}^{\infty }\hat{I}»(n)z^n$ is transcendental, where $\hat{I}»$ is Liouville's function. The transcendence of ${\sum }_{n=1}^{\infty }\hat{I}¼(n)z^n$ is also proved.

3 pages