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I am going through the article at this link, where the author proves that $\pi$ is $\text{transcendental}$ over $\mathbb{Q}$. Although, I understand the proof, I have some doubts.

  • At page $6$, the author defines a new function $f(x)$ as $f(x) = c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ Can anyone tell me what is the motivation behind defining $f(x)$ in this manner.
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    [Crossposted to MO](http://mathoverflow.net/questions/7817$0$/idea-behind-choosing-small-fx-as-csxp-1-frac-thetaxpp-1).2011-10-15

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Suppose that you want to show that some real number $x$ is transcendental. There is a standard way for going at this. This is introducing a so called "auxiliary function". In that paper, I think the auxiliary function is $F(x)$. You prove that, under the assumption that $x$ is algebraic, you show, for example, that $F(x)$ is positive and has a zero. (I didn't look what they actually do in the paper you quote.) The point is to derive a contradiction this way. All the proofs in transcendence theory (that I know) are done this way.

Finding the (or should I say "an") auxiliary polynomial is more of a puzzle actually.

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    I didn't know. Thnx!2011-10-15