Transcendence of power series for some number theoretic functions
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- by Peter Borwein and Michael Coons
- Proc. Amer. Math. Soc. 137 (2009), 1303-1305
- DOI: https://doi.org/10.1090/S0002-9939-08-09737-2
- Published electronically: October 28, 2008
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.References
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Bibliographic 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
- MR Author ID: 857151
- Email: mcoons@sfu.ca
- 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
- © Copyright 2008 By the authors
- Journal: Proc. Amer. Math. Soc. 137 (2009), 1303-1305
- MSC (2000): Primary 11J81, 11J99; Secondary 30B10, 26C15
- DOI: https://doi.org/10.1090/S0002-9939-08-09737-2
- MathSciNet review: 2465652