... 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:

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.