Transcendence of Power Series for Some Number Theoretic Functions

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dc.creatorCoons, Michael
dc.creatorBorwein, Peter
dc.date2008-06-10
dc.date.accessioned2026-07-07T09:43:34Z
dc.date.available2026-07-07T09:43:34Z
dc.descriptionWe 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, $\sum_{n=1}^\infty λ(n)z^n$ is transcendental, where $λ$ is Liouville's function. The transcendence of $\sum_{n=1}^\infty μ(n)z^n$ is also proved.
dc.description3 pages
dc.identifierhttps://arxiv.org/abs/0806.1563
dc.identifierhttp://arxiv.org/abs/0806.1563
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/162605
dc.subjectNumber Theory
dc.subject11J81; 11J99
dc.titleTranscendence of Power Series for Some Number Theoretic Functions
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