Transcendence of generating functions whose coefficients are multiplicative

Title: Transcendence of generating functions whose coefficients are multiplicative
Creator: Bell, Jason P.; Bruin, Nils; Coons, Michael
Relation: Transactions of the American Mathematical Society Vol. 364, Issue 2, p. 933-959
Publisher Link: http://dx.doi.org/10.1090/S0002-9947-2011-05479-6
Publisher: American Mathematical Society
Resource Type: journal article
Date: 2012
Description: In this paper, we give a new proof and an extension of the following result of Bézivin. Let f : ℕ → K be a multiplicative function taking values in a field K of characteristic 0, and write F(z) = Σ_n≥1f(n)zⁿ ∈ K[[z]] for its generating series. If F(z) is algebraic, then either there is a natural number k and a periodic multiplicative function χ(n) such that f(n) = n^kχ(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 = ℂ, we also prove that if the generating series of a multiplicative function is D-finite, then it must either be transcendental or rational.
Subject: algebraic functions; multiplicative functions; automatic sequences
Identifier: http://hdl.handle.net/1959.13/1354725
Identifier: uon:31335
Identifier: ISSN:0002-9947
Language: eng
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