Okay, I've been compiling a bunch of graphic information. And if we call,
\(
F_\lambda(s) = \lim_{n\to\infty} \log^{\circ n}\beta_\lambda(s+x_\lambda+n)\\
F_\lambda(0) = 1\,\,\text{for all}\,\, \Re(\lambda)>0\\
F_\lambda(s+1) = \exp(F_\lambda(s))\\
F_\lambda(s) : \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+\\
\)
Then,
\(
\text{tet}_\beta(s) = \lim_{\lambda\to 0^+} F_\lambda(s)\\
\)
What this is essentially saying is that we can interchange the limits. The construction I had was,
\(
\text{tet}_\beta(s) = \lim_{n\to\infty} \lim_{\lambda \to 0^+} \beta_\lambda(s+n+x_\lambda)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-\epsilon})\,\,\text{for}\,\,0 < \epsilon < 1\\
\)
But we can do one better by writing,
\(
\text{tet}_\beta(s) =\lim_{n\to\infty} \lim_{\lambda \to 0^+} \beta_\lambda(s+n+x_\lambda)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-\epsilon})\,\,\text{for}\,\,0 < \epsilon < 1\\
= \lim_{\lambda \to 0^+}\lim_{n\to\infty} \beta_\lambda(s+n+x_\lambda)
= \lim_{\lambda \to 0^+} F_\lambda(s)\\
\)
And that my construction was existence/construction; but once that's shown--we can use alternative ways of representing. We can just limit the multiplier to zero. We don't need anything too fancy.
This is to entice the reader into thinking that we are really just expanding the strip of holomorphy. Where,
\(
F_1(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.5}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 2\pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.25}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 4\pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.125}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 8 \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.0625}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 16 \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
\vdots\\
\text{tet}_\beta(z)\,\,\text{is holomorphic for}\,\,z\in\mathbb{C}\,\,\text{excluding}\,\,(-\infty,-2]\\
\)
I think this is the revelation I've been looking for. I'm able to make graphs of say \( F_{0.1}(z) \) much better than I can make a graph of \( \lim_{n\to\infty} \log^{\circ n} \beta(z+n) \); however, on the real line they barely disagree--about an error of 1E-10. So, as I see it; this cracks the code of how to map the Riemann sphere so that \( \lim_{\Im z \to \infty} \text{tet}_\beta(z) = \infty \). We've literally just shrunk the multiplier to zero.
So the periodic function \( \lambda(z) \) is actually just \( \lambda = 0 \); how cool is that!?!?!?!?
Guys, this is the cracked code of the beta method! The trivial solution to the equation,
\(
\text{tet}_\beta(z) = F_{\lambda(z)}(z)\\
\lambda(z+1) = \lambda(z)\\
\lim_{\Im z \to \infty} \lambda(z) = 0\\
\)
IS THE CORRECT SOLUTION! I can't believe I was always "normalizing" at the end. The trick is to include the function \( x_\lambda \) in the construction. All I have to do is show that the interchange of limits is perfectly legal; which isn't much. It's just ensuring on compact sets the limit stays stable. I'm so god damn excited! Can't believe I didn't think of this !
Regards, James
Here's a graph of \( F_{\log(1.5)}(z) \) over the domain \( -2 \le \Re(z) \le 3 \) and \( 0 \le \Im(z) \le 5 \); which is almost \( \text{tet}_\beta \); there's about a \( 1 E -3 \) discrepancy on the real line. This function is holomorphic for about \( |\Im(z)| \le 7 \) excluding the branch cut.
\(
F_\lambda(s) = \lim_{n\to\infty} \log^{\circ n}\beta_\lambda(s+x_\lambda+n)\\
F_\lambda(0) = 1\,\,\text{for all}\,\, \Re(\lambda)>0\\
F_\lambda(s+1) = \exp(F_\lambda(s))\\
F_\lambda(s) : \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+\\
\)
Then,
\(
\text{tet}_\beta(s) = \lim_{\lambda\to 0^+} F_\lambda(s)\\
\)
What this is essentially saying is that we can interchange the limits. The construction I had was,
\(
\text{tet}_\beta(s) = \lim_{n\to\infty} \lim_{\lambda \to 0^+} \beta_\lambda(s+n+x_\lambda)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-\epsilon})\,\,\text{for}\,\,0 < \epsilon < 1\\
\)
But we can do one better by writing,
\(
\text{tet}_\beta(s) =\lim_{n\to\infty} \lim_{\lambda \to 0^+} \beta_\lambda(s+n+x_\lambda)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-\epsilon})\,\,\text{for}\,\,0 < \epsilon < 1\\
= \lim_{\lambda \to 0^+}\lim_{n\to\infty} \beta_\lambda(s+n+x_\lambda)
= \lim_{\lambda \to 0^+} F_\lambda(s)\\
\)
And that my construction was existence/construction; but once that's shown--we can use alternative ways of representing. We can just limit the multiplier to zero. We don't need anything too fancy.
This is to entice the reader into thinking that we are really just expanding the strip of holomorphy. Where,
\(
F_1(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.5}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 2\pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.25}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 4\pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.125}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 8 \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.0625}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 16 \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
\vdots\\
\text{tet}_\beta(z)\,\,\text{is holomorphic for}\,\,z\in\mathbb{C}\,\,\text{excluding}\,\,(-\infty,-2]\\
\)
I think this is the revelation I've been looking for. I'm able to make graphs of say \( F_{0.1}(z) \) much better than I can make a graph of \( \lim_{n\to\infty} \log^{\circ n} \beta(z+n) \); however, on the real line they barely disagree--about an error of 1E-10. So, as I see it; this cracks the code of how to map the Riemann sphere so that \( \lim_{\Im z \to \infty} \text{tet}_\beta(z) = \infty \). We've literally just shrunk the multiplier to zero.
So the periodic function \( \lambda(z) \) is actually just \( \lambda = 0 \); how cool is that!?!?!?!?
Guys, this is the cracked code of the beta method! The trivial solution to the equation,
\(
\text{tet}_\beta(z) = F_{\lambda(z)}(z)\\
\lambda(z+1) = \lambda(z)\\
\lim_{\Im z \to \infty} \lambda(z) = 0\\
\)
IS THE CORRECT SOLUTION! I can't believe I was always "normalizing" at the end. The trick is to include the function \( x_\lambda \) in the construction. All I have to do is show that the interchange of limits is perfectly legal; which isn't much. It's just ensuring on compact sets the limit stays stable. I'm so god damn excited! Can't believe I didn't think of this !
Regards, James
Here's a graph of \( F_{\log(1.5)}(z) \) over the domain \( -2 \le \Re(z) \le 3 \) and \( 0 \le \Im(z) \le 5 \); which is almost \( \text{tet}_\beta \); there's about a \( 1 E -3 \) discrepancy on the real line. This function is holomorphic for about \( |\Im(z)| \le 7 \) excluding the branch cut.