... namely, in the seventh page of his Transcendental numbers & diophantine approximations. I do hope that somebody in the site has read this very paper recently... Here is a screenshot of the said page in the article under consideration:

see screenshot

Almost at the end of the second paragraph, Lang ascertains the existence of a largest integer $s$ such that $F(k\cdot z) = 0$, for all $k$ with $1 \leq k_{\nu} \leq s$. This allows me to state my first question:

Can anybody tell me how it is that he infers it from what he has mentioned previously?

It would seem that after picking the largest such $s$, he was to "fix" it. Yet, in one of the last lines in that page of the article we find, between parentheses, the following phrase: "for $s$ large".

How are we supposed to reconcile this paragraph with the one in which he actually defines $s$ as the largest integer such that $F(k\cdot z) = 0$, for all $k$ with $1 \leq k_{\nu} \leq s$.

While we are at it, is it correct the assertion in Murty & Rath's text on transcendental numbers (page 29) to the effect that a estimation of the type $\log |F(w)| \ll s^{5/2}$ implies that $|F(w)| \geq C^{-s^{5/2}}$ for some $C>0$?

Thanks in advance for your knowledgeable replies.

  • 1
    $\begingroup$ Lang constructs using Siegel's lemma (which is essentially a Dirichlet principle) a function with at least $n^3$ roots. So $s\geq n$. Now $n$ (and so $s$) can be arbitrary large. This is all very standard. You may try to read his book (Introduction to transcendental numbers) for slightly more details. $\endgroup$ Nov 15, 2016 at 14:48
  • 2
    $\begingroup$ To your last question: $\log |F(w)| \ll s^{5/2}$ means, by definition, that for some constant $K>0$ we have $-Ks^{5/2}\leq\log |F(w)|\leq Ks^{5/2}$, i.e., $C^{-s^{5/2}}\leq|F(w)|\leq C^{s^{5/2}}$ with $C:=e^K$ (note that $C>1$). Hope this helps. $\endgroup$
    – GH from MO
    Nov 15, 2016 at 20:21


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