- 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) = Σn≥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) = nkχ(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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