Transcendence of Power Series for Some Number Theoretic Functions

Details Transcendence of Power Series for Some Number Theoretic Functions Transcendence of Power Series for Some Number Theoretic Functions

2008-06-10

arxiv.org/abs/0806.1563v1

Mathematics

Number Theory

3 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Transcendence of Power Series for Some Number Theoretic Functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Transcendence of Power Series for Some Number Theoretic Functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Transcendence of Power Series for Some Number Theoretic Functions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-174408