Transcendence of power series for some number theoretic functions

Abstract:

We give a new proof of Fatou’s theorem: 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.