08/18/2022, 04:18 AM
Hey, tommy, so essentially I glossed over the details there on "converges pointwise." It's very hard to describe.
First of all the beta function itself is a holomorphic function. And the iteration \(\log^{\circ n} \beta(s+n)\) does converge almost everywhere in \(\mathbb{C}\). But the taylor data does not converge.
It's pretty hard to explain how this happens, and the best I can do is point you to my arxiv paper describing it.
But essentially:
\[
\lim_{n\to\infty} \log^{\circ n} \beta(s+n) = \sum_{n=0}^\infty a_n (s-s_0)^n\\
\]
Is divergent, and at best creates an asymptotic series. Which satisfies the functional equation.
You are correct about your statements of bounded continuously differentiable (locally biholomorphic); but this object doesn't converge to a bounded continuously differentiable function. (We get something like a devil's staircase, but in the complex plane).
That out of the way, the beta method is holomorphic when the base \(b = e^{\mu}\) is within Shell thron. And it is also holomorphic for some \(b = e^{\mu}\) outside of shell thron. But when we use the beta method for \(b > \eta\), it's nowhere holomorphic. This is largely because you can show that:
\[
\log^{\circ n} \beta(s+n)\\
\]
Converges. BUT! for any compact set \(N\) about a point \(s_0\):
\[
\log^{\circ n} \beta(s+n)\,\,\text{does not converge uniformly on}\,N\\
\]
And this forces us to not have holomorphy. Despite getting a pointwise value (which doesn't happen all the time, but happens almost everywhere in \(\mathbb{C}\)). So it really is quite the anomaly when \(b > \eta\).
Also note, this doesn't apply to the gaussian method you propose. I'm pretty sure this can be massaged to converge to Kneser. So it's not that this is never holomorphic. Just when we use the logistic function, it isn't. So if we try to make a \(2 \pi i / \lambda\) periodic tetration base \(b=e\)--the best we can do is make asymptotic solutions. Which means that \(\log^{\circ n} \beta(s+n)\) is littered with branch cuts for fixed \(n\), and they become more dense as we increase \(n\). But point wise, it still converges to a value. And we can uncover an asymptotic series. It's pretty hard to explain. I spent about 90 pages trying to explain how and when we derive holomorphy for the beta method. So I can't really cover it here.
A lot of the work was done by Sheldon and myself, and there are still threads on here where sheldon argues pretty strongly that \(2 \pi i\) periodic \(b = e\) tetration is only smooth on \(\mathbb{R}\) and the taylor data doesn't converge. So what you get are asymptotic expansions at everypoint. We get something similar in the complex plane, asymptotic expansions; The taylor series at a point diverges.
First of all the beta function itself is a holomorphic function. And the iteration \(\log^{\circ n} \beta(s+n)\) does converge almost everywhere in \(\mathbb{C}\). But the taylor data does not converge.
It's pretty hard to explain how this happens, and the best I can do is point you to my arxiv paper describing it.
But essentially:
\[
\lim_{n\to\infty} \log^{\circ n} \beta(s+n) = \sum_{n=0}^\infty a_n (s-s_0)^n\\
\]
Is divergent, and at best creates an asymptotic series. Which satisfies the functional equation.
You are correct about your statements of bounded continuously differentiable (locally biholomorphic); but this object doesn't converge to a bounded continuously differentiable function. (We get something like a devil's staircase, but in the complex plane).
That out of the way, the beta method is holomorphic when the base \(b = e^{\mu}\) is within Shell thron. And it is also holomorphic for some \(b = e^{\mu}\) outside of shell thron. But when we use the beta method for \(b > \eta\), it's nowhere holomorphic. This is largely because you can show that:
\[
\log^{\circ n} \beta(s+n)\\
\]
Converges. BUT! for any compact set \(N\) about a point \(s_0\):
\[
\log^{\circ n} \beta(s+n)\,\,\text{does not converge uniformly on}\,N\\
\]
And this forces us to not have holomorphy. Despite getting a pointwise value (which doesn't happen all the time, but happens almost everywhere in \(\mathbb{C}\)). So it really is quite the anomaly when \(b > \eta\).
Also note, this doesn't apply to the gaussian method you propose. I'm pretty sure this can be massaged to converge to Kneser. So it's not that this is never holomorphic. Just when we use the logistic function, it isn't. So if we try to make a \(2 \pi i / \lambda\) periodic tetration base \(b=e\)--the best we can do is make asymptotic solutions. Which means that \(\log^{\circ n} \beta(s+n)\) is littered with branch cuts for fixed \(n\), and they become more dense as we increase \(n\). But point wise, it still converges to a value. And we can uncover an asymptotic series. It's pretty hard to explain. I spent about 90 pages trying to explain how and when we derive holomorphy for the beta method. So I can't really cover it here.
A lot of the work was done by Sheldon and myself, and there are still threads on here where sheldon argues pretty strongly that \(2 \pi i\) periodic \(b = e\) tetration is only smooth on \(\mathbb{R}\) and the taylor data doesn't converge. So what you get are asymptotic expansions at everypoint. We get something similar in the complex plane, asymptotic expansions; The taylor series at a point diverges.
