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
N
\mathbb {N}
to
{
−
1
,
1
}
\{-1,1\}
, the series
∑
n
=
1
∞
f
(
n
)
z
n
\sum _{n=1}^\infty f(n)z^n
is transcendental over
Z
(
z
)
\mathbb {Z}(z)
; in particular,
∑
n
=
1
∞
λ
(
n
)
z
n
\sum _{n=1}^\infty \lambda (n)z^n
is transcendental, where
λ
\lambda
is Liouville’s function. The transcendence of
∑
n
=
1
∞
μ
(
n
)
z
n
\sum _{n=1}^\infty \mu (n)z^n
is also proved.