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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Transcendence of power series for some number theoretic functions


Authors: Peter Borwein and Michael Coons
Journal: Proc. Amer. Math. Soc. 137 (2009), 1303-1305
MSC (2000): Primary 11J81, 11J99; Secondary 30B10, 26C15
Published electronically: October 28, 2008
MathSciNet review: 2465652
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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.


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Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.sfu.ca

Michael Coons
Affiliation: Department of Mathematics, Simon Fraser University, British Columbia, Canada V5A 1S6
Email: mcoons@sfu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09737-2
PII: S 0002-9939(08)09737-2
Received by editor(s): May 30, 2008
Published electronically: October 28, 2008
Additional Notes: Research supported in part by grants from NSERC of Canada and MITACS
Communicated by: Ken Ono
Article copyright: © Copyright 2008 By the authors