Hey, Everyone; been a long time...

[sheldonison]

HEY SHELDON! Long time no talk! I'm excited to discuss. Just because I'm being a little prideful, \( \phi(x) \) is NOT a superfunction. Not in any way shape or form. It's more like a superfunction, but with an exponential corrective term.

This is to mean that,

\(
\phi(s) = e^{\displaystyle s-1 + e^{s-2+e^{s-3....}}}
\)

So that,

\( \phi(s+1) = e^{s+\phi(s)} \)

(Believe it or not, this function is ENTIRE!)

Which I'm sure you can astutely note is NOT an inverted Abel function. The entire paper begins with constructing this function (which is the real novelty of the technique); and then, that old tried method that failed so many times, every time I read about it, finally works. Which is to say,

\(
\lim_{n\to\infty} \log \log \cdots (n\,\text{times})\cdots \log \phi(s+n) = e \uparrow \uparrow s + \omega
\)

(Which although it looks like Tommy's technique; or has wafts of Kouznetsov; it's absolutely more convenient to use \( \phi \).)
...

James,
Looks like I have some catching up to do.  I haven't gone back to your paper yet, because I wanted to think about \( \phi(s) \) on my own more first.  I wanted to generate a formal power series for f which leads tor \phi.  I started to generate the formal series for f, so I am convinced enough that there is a unique definition for \( \phi \) that is indeed well defined and entire and is \( 2\pi i \) periodic.
 Back to real work, and then back to your paper when I am able.  Again, thanks for posting James, it is a delight to see some well thought out results from your work.
\( f=x+\sum_{n=2}^{\infty}a_n\cdot\.x^n\;\;\;\phi(s)=f(e^s)\;\;\;\phi(s+1) = e^{s+\phi(s)} \)

So for Re(z) negative enough, psi(z) looks more or less like exp(z), and as z at the real axis increases, psi(z) looks like a super exponential.   Psi is 2 pi i periodic.  

So how does psi(z) behave at Im(z)=pi*i as the real(z) increases?  When Im(z)=Pi*i, I think it is always 0>psi(z)>abs(exp(z), but psi will eventually oscillate between negative values approaching the magnitude of exp(z), and nearly zero values ... 

Also, I'm curious to is if phi might lead to an analytic solution for tet_e(x), instead of one only defined at the real axis, which is typically what we see for Tommy's iterated logarithm approach.