08/17/2009, 08:50 AM
Ya perhaps that line of thought will not contribute to our topic.
My concern was about the difference of points just somehow converging to the real axis, compared with that they indeed come close to every single point of the real axis.
But this seems to be true.
To properly verify this we need to show that for each \( x>0 \) and for every \( \varepsilon>0 \) there exist natural numbers \( n \) and \( k \) such that \( \left|x-\log_b^{[n-3]}(\log_b^{[3]}(a)+2\pi i k)\right|<\varepsilon \).
However did we silently switch from base change \( \eta\to e \) (\( b=\eta,a=e \)) to the base change \( e\to\eta \) (\( b=e,a=\eta \))?
I think we are more interested in the first one!
My concern was about the difference of points just somehow converging to the real axis, compared with that they indeed come close to every single point of the real axis.
But this seems to be true.
To properly verify this we need to show that for each \( x>0 \) and for every \( \varepsilon>0 \) there exist natural numbers \( n \) and \( k \) such that \( \left|x-\log_b^{[n-3]}(\log_b^{[3]}(a)+2\pi i k)\right|<\varepsilon \).
However did we silently switch from base change \( \eta\to e \) (\( b=\eta,a=e \)) to the base change \( e\to\eta \) (\( b=e,a=\eta \))?
I think we are more interested in the first one!