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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Transcendence of generating functions whose coefficients are multiplicative
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by Jason P. Bell, Nils Bruin and Michael Coons PDF
Trans. Amer. Math. Soc. 364 (2012), 933-959 Request permission

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.
References
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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
  • MR Author ID: 632303
  • Email: jpb@sfu.ca
  • Nils Bruin
  • Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6
  • MR Author ID: 653028
  • Email: nbruin@sfu.ca
  • Michael Coons
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
  • MR Author ID: 857151
  • Email: mcoons@math.uwaterloo.ca
  • 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2846359