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Transcendence of generating functions whose coefficients are multiplicative


Authors: Jason P. Bell, Nils Bruin and Michael Coons
Journal: Trans. Amer. Math. Soc. 364 (2012), 933-959
MSC (2010): Primary 11N64, 11J91; Secondary 11B85
DOI: https://doi.org/10.1090/S0002-9947-2011-05479-6
Published electronically: August 31, 2011
MathSciNet review: 2846359
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Abstract: In this paper, we give a new proof and an extension of the following result of Bézivin. Let $ f:\mathbb{N}\to K$ be a multiplicative function taking values in a field $ K$ of characteristic 0, and write $ F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for its generating series. If $ F(z)$ is algebraic, then either there is a natural number $ k$ and a periodic multiplicative function $ \chi(n)$ such that $ f(n)=n^k \chi(n)$ for all $ n$ or $ f(n)$ is eventually zero. In particular, the generating series of a multiplicative function taking values in a field of characteristic zero is either transcendental or rational. For $ K=\mathbb{C}$, we also prove that if the generating series of a multiplicative function is $ D$-finite, then it must either be transcendental or rational.


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Additional Information

Jason P. Bell
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6
Email: jpb@sfu.ca

Nils Bruin
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6
Email: nbruin@sfu.ca

Michael Coons
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: mcoons@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9947-2011-05479-6
Keywords: Algebraic functions, multiplicative functions, automatic sequences
Received by editor(s): March 15, 2010
Received by editor(s) in revised form: August 27, 2010
Published electronically: August 31, 2011
Additional Notes: The research of the first and second authors was supported in part by a grant from NSERC of Canada
The research of the third author was supported by a Fields-Ontario Fellowship.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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