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Auxiliary Polynomials in Number Theory by David Masser

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3

Irrationality II – Mahler’s Method

Here we prove an analogue of Theorem 2.1 for the function

image

which was considered by Mahler. Now this series converges in the complex plane only on the unit disc D defined by |z| < 1; and indeed it has a so-called natural boundary on |z| = 1 which precludes any further analytic continuation. Near z = 0 it converges rather more rapidly than image and so we might expect things to be easier here. So let us again see what truncation of f(α) gives.

Let us start with α = a/b, for simplicity taking a ≥ 1, b ≥ 1; but now a

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