(02/03/2021, 11:44 PM)tommy1729 Wrote: (02/02/2021, 04:40 AM)JmsNxn Wrote: Hey, Tommy!
So I had noticed that you implicitly assumed \( V(s+1) = V(s) \) in your original analysis. I didn't notice it right away, but I noticed it afterwards when I saw it would imply \( 2 \pi i \) periodicity. I looked at it some more, and was pretty sure you were on to something--but never been too good with the Lambert function. This makes much much more sense quite frankly. Especially if we think of the branch cuts appearing at \( \Im(s) = 2\pi k \) for \( k \in \mathbb{Z} \). This is where our \( \phi \) function will recycle, and a cluster of singularities will force non-analycity of \( \tau \).
Now I am confused.
You say non-analytic here.
And you also wrote 2 papers claiming analytic ?
Im aware of Sheldon's arguments and the complexity of tetration.
But the point is I am confused about your viewpoint.
I mean non-analycity of \( \tau \) would imply non-analytic tetration right ?
But you have 2 papers claiming analyticity and intend to explain it further.
Regards
tommy1729
Hey, Tommy. I'll clarify my stance. I initially thought I had showed that,
\(
e \uparrow \uparrow s : \mathbb{C} / \mathcal{L} \to \mathbb{C}\\
\)
Is a holomorphic function upto a nowhere dense set \( \mathcal{L} \). Now this, I believe is technically correct, but I had implicitly assumed that it is analytic on \( \mathbb{R}^+ \). Sheldon, thoroughly convinced me that this probably doesn't happen. What I believe now, which is essentially the above statement, except,
\(
e \uparrow \uparrow s: \mathbb{C} / (\mathbb{R}+2\pi i k) \to \mathbb{C}\\
\)
Which explicitly states where it is holomorphic. This is to say, it is still holomorphic upto a nowhere dense set; but \( \mathbb{R}^+ \) seems to be in this set. The mistake I made was pretty foolhardy,
I had assumed that,
\(
\frac{d}{dy}|\phi(t+iy)| = 0\,\,\text{iff}\,\,y = k\pi\\
\)
so that \( |\phi(t+\pi i)| \) is a global minimum. But this isn't so. What I believe I've shown now, is that it is only a local minimum, but the domain in which it is a minimum eventually grows to \( \delta < y < 2\pi - \delta \) for large enough \( t\ge T \)--and from here the paper continues as it did before with the construction of \( \tau \). The problem being,
\(
\frac{d}{dy}|_{y=y_k} |\phi(t+iy)| = 0\\
\)
Has solutions \( y_k \) which cluster towards \( \mathbb{R}^+ \) as \( t\to \infty \). This causes our function \( |\phi(s)| \) to dip towards small values, causing \( \log \log ... \log \phi(s+n) \) to hit a singularity. Essentially we hit a wall of singularities at the real line. But in the strip \( 0 < \Im(s) < 2\pi \) we have no such problem because \( |\phi(t+\pi i)| \) grows and acts like a minimum in the strip \( \delta < \Im(s) < 2\pi - \delta \); forcing our construction of \( \tau \) to converge.
All in all; I was incorrect to think I showed analycity on \( \mathbb{R}^+ \)--but I do believe this is still holomorphic; just unfortunately not for real values. At best I can show is continuously differentiable, but I don't think a \( C^{\infty} \) proof is that out of reach.
All in all I was half-right at best. Also, the second paper, is just the same paper accounting for this foolhardy mistake--and trying to correct it--and state a stronger version of what I had originally stated.
