Transcendence of power series for some number theoretic functions
Peter Borwein, Michael James Coons · Proceedings of the American Mathematical Society · 2008
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 N \mathbb {N} to { − 1 , 1 } \{-1,1\} , the series ∑ n = 1 ∞ f ( n ) z n \sum _{n=1}^\infty f(n)z^n is transcendental over Z ( z ) \mathbb {Z}(z) ; in particular, ∑ n = 1 ∞ λ ( n ) z n \sum _{n=1}^\infty \lambda (n)z^n is transcendental, where λ \lambda is Liouville’s function. The transcendence of ∑ n = 1 ∞ μ ( n ) z n \sum _{n=1}^\infty \mu (n)z^n is also proved.