**Here's the arXiv link with everything**
https://arxiv.org/abs/2104.01990
Hey everyone,
I've spent the past nine days hammering out a construction of tetration. This is everything I wanted in \( \phi \) (to such an extent that many of the proof layouts I had for \( \phi \) could be transferred over). The main focus of this construction is a family of functions \( \beta_\lambda(s) \) which satisfy \( \log\beta_\lambda(s+1) = \beta_\lambda(s) + \mathcal{O}(e^{-\lambda s}) \) as \( \Re(\lambda s) \to \infty \).
I worked my ass off on this for nine days, because I'm 99.99999% sure I'm correct. I'm scared there's some tiny flaw I'm not noticing. But, nonetheless, I'm happy to post this here; even more confidently than the \( \phi \) method, while using much of the same principles. I've eliminated all the errors and potential errors from the \( \phi \) method.
I'm confident I have constructed a holomorphic tetration function \( \text{tet}_\beta(s) : \mathbb{C}/(-\infty,-2] \to \mathbb{C} \) which is real-valued. I don't know anything about its behaviour at \( \Re(s) = -\infty,\,\Im(s) > 0 \) and no idea of its behaviour at \( \Im(s) \to \infty \). I do think that as \( \Re(s) \to -\infty \) is equivalent to either the julia set or fatou set of \( \log \); and this is not a uniform convergence to a fixed point, like with Kneser. I'm pretty certain that as \( \Im(s) \to \infty \) we'll get that \( \text{tet}_\beta(s) \) should oscillate wildly and behave like orbits of \( e^z \)--eventually hitting infinity and back to 0.
I don't know how to summarize this whole construction, but I believe I've done everything I had planned with \( \phi \); I just mis-stepped in assuming I only need a single function rather than a family. And this family of functions needed to approximate the solution of tetration at infinity; and do so in a uniform manner.
...
A special thanks goes to tommy for the weird infinite compositions he played with...
...
And to Sheldon for running the numbers on \( \phi \)...
...
If you want to read this paper; this requires a lot of infinite compositions, and brushes on Riemann surfaces, and a lot (I mean a lot) of complex function theory.
Regards, James
Here is the final update on this paper, and its final iteration.
FINAL TETRATION ASYMPTOTIC.pdf (Size: 3.66 MB / Downloads: 832)
Here is the original version of the paper, which still had some errors
asymptotic_tetration__2.pdf (Size: 375.8 KB / Downloads: 859)
UPDATE!!
So I'm working on an update to the above paper. The things I have changed and the things I am planning to change:
1. I bettered the proof that \( \beta_\lambda(s) \to \infty \) as \( \Re(s) \to \infty \). I spend more time explaining this, as I realize the entire paper hinges on this. And it's exactly why \( \beta_\lambda(s) \) is superior to \( \phi(s) \).
2. I realized \( \tau_\lambda^n(s) \) isn't an exponential series--but it looks enough like one for it not to affect the final result. I went a bit quick here in the first iteration--because I knew it didn't matter if it was either way (but I forgot to double check). The correct asymptotic is \( \tau_\lambda^n(s) = - \log(1+e^{-\lambda s}) + o(e^{-\lambda s}) \). The second iteration of this paper follows similarly; except when talking about convergence we have to be more careful with our compact sets. The functions \( u_\lambda^n(w) \) are not holomorphic for \( |w| \le \delta \), they're holomorphic for \( \{0 < |w| \le \delta,\,w \neq -e^{-\lambda j}, j\ge 1\} \). But! The singularity at \( 0 \) is removable in a specific manner--\( u_\lambda^n(e^{-\lambda j}w)/e^{-\lambda j}w \to -1 \) as \( j\to\infty \). From which, most of the paper isn't really changed, it's just a bit more aggressive (in that there are more arguments). I've finished all this.
3. I'm trying to visualize some of the constructs. And I plan to include some graphics to help me explain some of these things--especially the domain arguments. I've already written out some functor diagrams I'm including. I'm horrible with graphics programming, though--so, it'll take me a while to develop some nice computer produced \( x,y \)-plane type graphs.
4. I clarified a lot of the language--at least part way. I haven't fixed everything yet.
5. I'm working on trying to develop a Taylor series at zero for computational purposes, but I lack far too much computational knowledge (especially because pari-gp is too unfamiliar to me, and the only language I ever really worked with was C and python). Everything I do just hits overflow before I'm even at \( \beta_\lambda \) for large arguments. Which says more about my coding than my method...
Again, the main result hasn't changed at all. Just how we get there is a tad more difficult; but I'm explaining it much better. If you felt confused from the first iteration, I apologize. I'll clear everything up. I got ahead of myself--again, I wrote that in 9 days in a light-bulb moment. The proof schema is still the exact same; just some of the brickwork needs to be patched, is all.
Regards, James.
UPDATE! #2
Please see post #3 for all the information I changed in this update. I, about, doubled the length of this paper. But it's much more solid now.
asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 838)
UPDATE #3
Please see post #4 for all the information I changed in this update. This is the final update, and as far as I see, constructs a holomorphic tetration. I added about 19 graphs, and I think that's the most I can do. I think I got out everything that needed to be shown. It's still a little rough around the edges, but I believe this constructs tetration. And further, constructs it pretty well.
FINAL TETRATION ASYMPTOTIC.pdf (Size: 3.66 MB / Downloads: 832)
https://arxiv.org/abs/2104.01990
Hey everyone,
I've spent the past nine days hammering out a construction of tetration. This is everything I wanted in \( \phi \) (to such an extent that many of the proof layouts I had for \( \phi \) could be transferred over). The main focus of this construction is a family of functions \( \beta_\lambda(s) \) which satisfy \( \log\beta_\lambda(s+1) = \beta_\lambda(s) + \mathcal{O}(e^{-\lambda s}) \) as \( \Re(\lambda s) \to \infty \).
I worked my ass off on this for nine days, because I'm 99.99999% sure I'm correct. I'm scared there's some tiny flaw I'm not noticing. But, nonetheless, I'm happy to post this here; even more confidently than the \( \phi \) method, while using much of the same principles. I've eliminated all the errors and potential errors from the \( \phi \) method.
I'm confident I have constructed a holomorphic tetration function \( \text{tet}_\beta(s) : \mathbb{C}/(-\infty,-2] \to \mathbb{C} \) which is real-valued. I don't know anything about its behaviour at \( \Re(s) = -\infty,\,\Im(s) > 0 \) and no idea of its behaviour at \( \Im(s) \to \infty \). I do think that as \( \Re(s) \to -\infty \) is equivalent to either the julia set or fatou set of \( \log \); and this is not a uniform convergence to a fixed point, like with Kneser. I'm pretty certain that as \( \Im(s) \to \infty \) we'll get that \( \text{tet}_\beta(s) \) should oscillate wildly and behave like orbits of \( e^z \)--eventually hitting infinity and back to 0.
I don't know how to summarize this whole construction, but I believe I've done everything I had planned with \( \phi \); I just mis-stepped in assuming I only need a single function rather than a family. And this family of functions needed to approximate the solution of tetration at infinity; and do so in a uniform manner.
...
A special thanks goes to tommy for the weird infinite compositions he played with...
...
And to Sheldon for running the numbers on \( \phi \)...
...
If you want to read this paper; this requires a lot of infinite compositions, and brushes on Riemann surfaces, and a lot (I mean a lot) of complex function theory.
Regards, James
Here is the final update on this paper, and its final iteration.
FINAL TETRATION ASYMPTOTIC.pdf (Size: 3.66 MB / Downloads: 832)
Here is the original version of the paper, which still had some errors
asymptotic_tetration__2.pdf (Size: 375.8 KB / Downloads: 859)
UPDATE!!
So I'm working on an update to the above paper. The things I have changed and the things I am planning to change:
1. I bettered the proof that \( \beta_\lambda(s) \to \infty \) as \( \Re(s) \to \infty \). I spend more time explaining this, as I realize the entire paper hinges on this. And it's exactly why \( \beta_\lambda(s) \) is superior to \( \phi(s) \).
2. I realized \( \tau_\lambda^n(s) \) isn't an exponential series--but it looks enough like one for it not to affect the final result. I went a bit quick here in the first iteration--because I knew it didn't matter if it was either way (but I forgot to double check). The correct asymptotic is \( \tau_\lambda^n(s) = - \log(1+e^{-\lambda s}) + o(e^{-\lambda s}) \). The second iteration of this paper follows similarly; except when talking about convergence we have to be more careful with our compact sets. The functions \( u_\lambda^n(w) \) are not holomorphic for \( |w| \le \delta \), they're holomorphic for \( \{0 < |w| \le \delta,\,w \neq -e^{-\lambda j}, j\ge 1\} \). But! The singularity at \( 0 \) is removable in a specific manner--\( u_\lambda^n(e^{-\lambda j}w)/e^{-\lambda j}w \to -1 \) as \( j\to\infty \). From which, most of the paper isn't really changed, it's just a bit more aggressive (in that there are more arguments). I've finished all this.
3. I'm trying to visualize some of the constructs. And I plan to include some graphics to help me explain some of these things--especially the domain arguments. I've already written out some functor diagrams I'm including. I'm horrible with graphics programming, though--so, it'll take me a while to develop some nice computer produced \( x,y \)-plane type graphs.
4. I clarified a lot of the language--at least part way. I haven't fixed everything yet.
5. I'm working on trying to develop a Taylor series at zero for computational purposes, but I lack far too much computational knowledge (especially because pari-gp is too unfamiliar to me, and the only language I ever really worked with was C and python). Everything I do just hits overflow before I'm even at \( \beta_\lambda \) for large arguments. Which says more about my coding than my method...
Again, the main result hasn't changed at all. Just how we get there is a tad more difficult; but I'm explaining it much better. If you felt confused from the first iteration, I apologize. I'll clear everything up. I got ahead of myself--again, I wrote that in 9 days in a light-bulb moment. The proof schema is still the exact same; just some of the brickwork needs to be patched, is all.
Regards, James.
UPDATE! #2
Please see post #3 for all the information I changed in this update. I, about, doubled the length of this paper. But it's much more solid now.
asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 838)
UPDATE #3
Please see post #4 for all the information I changed in this update. This is the final update, and as far as I see, constructs a holomorphic tetration. I added about 19 graphs, and I think that's the most I can do. I think I got out everything that needed to be shown. It's still a little rough around the edges, but I believe this constructs tetration. And further, constructs it pretty well.
FINAL TETRATION ASYMPTOTIC.pdf (Size: 3.66 MB / Downloads: 832)
