So, I reread my modular stuff, and I made a couple math typos in the above post, but the general idea is still there. Again, this is just undeveloped research. Take it all with a grain of salt. I'm doing an info dump....
As I realize I haven't explained what modular functions are, or do; I thought I'd remind/explain to everyone what I mean here.
\[
\phi(s) = \sum_{j=-k}^\infty a_k e^{2\pi i j s}
\]
This series converges uniformly on compact subsets of \(\Im(s) > 0\) upto some singularities; though you need to change \(a_k\) (varying where you center the Laurent expansion), and also converges at least uniformly on \([0,1]\) up to two discontinuities (for us at least). The key to a modular function has nothing to do with this though.
A modular function is invariant under a special group of automorphisms of \(\mathbb{H} = \{\Im(s) > 0\}\). So not only is it periodic, it also satisfies:
\[
\phi(\frac{az+b}{cz+d}) = \phi(z)\\
\]
Where \(ad-bc = 1\) for \(a,b \in \mathbb{Z}\).
We can say this about our \(\phi\) because it is invariant under the automorphism \(z \mapsto z+1\); and not under any \(z \mapsto z+c\) for \(c \in \mathbb{R}/\mathbb{Z}\); This produces the first generator of the special group of automorphisms of \(\mathbb{H}\), the modular \(\Gamma\) group. To get the second generator is a bit more complicated, and relates deeper to work I did in the paper. If you can derive that \(\phi(-1/z) = \phi(z)\), then \(\phi\) must be modular...
If I write \(h(s) = 1/\beta(s)\), then: \(h(s+1) = e^{-1/h(s)}(1+e^{i\phi(s)s})\)--notice that \(h\) uses the exact generator we need: \(-1/z\). This caries over to tetration too, \(H = 1/\text{tet}\), then \(H(s+1) = b^{-1/H(s)}\). This is oddly very important, and just helps us get a hang that the modular aspect isn't that far out there, considering the second generator makes a direct appearance with the multiplicative inverse.
To get there though, we have to analyze what we're asking of \(\phi\). We must have that \(\phi(s) \to \infty\) as \(\Im(s) \to \infty\) (look at the expansion). Additionally, as we want to relate this to the beta method, we have to address that we have surjectivity to \(\widehat{\mathbb{C}}\). First of all \(\beta(s) \to 0\) as \(|s| \to \infty\) for \(\arg(s) \ge \pi/2\); and further we'd like \(\tau(s) \to L\). (Remember, we are trying to reconstruct Kneser).
Exactly as in Tommy's Gaussian method, we want as \(|s| \to \infty\) while \(0\le \arg(s) \le \pi/2\) that \(\beta(s+1) = e^{\beta(s)}\). This means we want \(e^{i\phi(s)s} \to 0\), but since here, \(\phi(s) \to i\infty\) and \(i\phi(s)s \to - \infty\), we're all good. So in this area \(\beta(s+1) \approx e^{\beta(s)}\). But \(\phi:\mathbb{R}\cup\infty\to\mathbb{R}\cup\infty\) as well (it must map the boundary to itself). But additionally, this means it has poles on \([0,1]\), we can choose a pole such that \(\phi(\infty) \sim \phi(-0)\), and this gets us \(\phi(-1/s) \sim \phi(s)\) as \(|s| \to 0,\infty\) for \(s \in \mathbb{H}\). Ensure this for all the derivatives too, and \(\phi(s) = \phi(-1/s)\) by virtue of holomorphy.
This is enough for us to say that it fixes the automorphic group. The automorphic group \(\Gamma\) is generated by \(z\mapsto z+1,\,\,z\mapsto-1/z\).
...I might need a bit more, I'll have to reread all the modular books I've read but this is at least 90% there. There's probably something small you also need to derive that \(\phi\) must be modular, but I can't think of much off the top of my head. Again, I'm just spitballing areas of research I haven't fully developed.... All in all, we've asked for a lot of the modular function \(\phi(s)\), I'd bet money that the only one that works is unique (I know it's unique because Kneser is unique), and so there's some modular function that makes Kneser.
This would mean that \(\beta\) would have a flavor of modular functions; it would still admit singularities but the singularities would look like discontinuities between \(\beta(s+1) = e^{\beta(s)}\) or \(\beta(s+1) = 0\), based on how \(e^{i\phi(s)s} = 0,\infty\) when \(\phi\) has a singularity. Additionally you'd be able to talk fluidly about the transformation \(1/\beta(s)\) is equivalent to \(z \mapsto -1/z\) much more fluidly (again, this was a large part of the paper for describing the weak Julia set).
This would mean we wouldn't get "singularities" in \(\beta\), we would get essential singularities that look like \(\beta(s+1) = 0,\beta(s+1) = e^{\beta(s)}\). These, in the end would become our values of kneser that equal the orbit \(\exp^{n}(0)\). This shit is unbelievably fascinating. But I have so much ODE/PDE research I've put on hold for 2 years to look at tetration; I need a break, rofl.
So, I'm going back into my dark little corner of studying Schrodinger's equation, and how it relates to infinite compositions. See you guys on the otherside
!!!
Regards, James
As I realize I haven't explained what modular functions are, or do; I thought I'd remind/explain to everyone what I mean here.
\[
\phi(s) = \sum_{j=-k}^\infty a_k e^{2\pi i j s}
\]
This series converges uniformly on compact subsets of \(\Im(s) > 0\) upto some singularities; though you need to change \(a_k\) (varying where you center the Laurent expansion), and also converges at least uniformly on \([0,1]\) up to two discontinuities (for us at least). The key to a modular function has nothing to do with this though.
A modular function is invariant under a special group of automorphisms of \(\mathbb{H} = \{\Im(s) > 0\}\). So not only is it periodic, it also satisfies:
\[
\phi(\frac{az+b}{cz+d}) = \phi(z)\\
\]
Where \(ad-bc = 1\) for \(a,b \in \mathbb{Z}\).
We can say this about our \(\phi\) because it is invariant under the automorphism \(z \mapsto z+1\); and not under any \(z \mapsto z+c\) for \(c \in \mathbb{R}/\mathbb{Z}\); This produces the first generator of the special group of automorphisms of \(\mathbb{H}\), the modular \(\Gamma\) group. To get the second generator is a bit more complicated, and relates deeper to work I did in the paper. If you can derive that \(\phi(-1/z) = \phi(z)\), then \(\phi\) must be modular...
If I write \(h(s) = 1/\beta(s)\), then: \(h(s+1) = e^{-1/h(s)}(1+e^{i\phi(s)s})\)--notice that \(h\) uses the exact generator we need: \(-1/z\). This caries over to tetration too, \(H = 1/\text{tet}\), then \(H(s+1) = b^{-1/H(s)}\). This is oddly very important, and just helps us get a hang that the modular aspect isn't that far out there, considering the second generator makes a direct appearance with the multiplicative inverse.
To get there though, we have to analyze what we're asking of \(\phi\). We must have that \(\phi(s) \to \infty\) as \(\Im(s) \to \infty\) (look at the expansion). Additionally, as we want to relate this to the beta method, we have to address that we have surjectivity to \(\widehat{\mathbb{C}}\). First of all \(\beta(s) \to 0\) as \(|s| \to \infty\) for \(\arg(s) \ge \pi/2\); and further we'd like \(\tau(s) \to L\). (Remember, we are trying to reconstruct Kneser).
Exactly as in Tommy's Gaussian method, we want as \(|s| \to \infty\) while \(0\le \arg(s) \le \pi/2\) that \(\beta(s+1) = e^{\beta(s)}\). This means we want \(e^{i\phi(s)s} \to 0\), but since here, \(\phi(s) \to i\infty\) and \(i\phi(s)s \to - \infty\), we're all good. So in this area \(\beta(s+1) \approx e^{\beta(s)}\). But \(\phi:\mathbb{R}\cup\infty\to\mathbb{R}\cup\infty\) as well (it must map the boundary to itself). But additionally, this means it has poles on \([0,1]\), we can choose a pole such that \(\phi(\infty) \sim \phi(-0)\), and this gets us \(\phi(-1/s) \sim \phi(s)\) as \(|s| \to 0,\infty\) for \(s \in \mathbb{H}\). Ensure this for all the derivatives too, and \(\phi(s) = \phi(-1/s)\) by virtue of holomorphy.
This is enough for us to say that it fixes the automorphic group. The automorphic group \(\Gamma\) is generated by \(z\mapsto z+1,\,\,z\mapsto-1/z\).
...I might need a bit more, I'll have to reread all the modular books I've read but this is at least 90% there. There's probably something small you also need to derive that \(\phi\) must be modular, but I can't think of much off the top of my head. Again, I'm just spitballing areas of research I haven't fully developed.... All in all, we've asked for a lot of the modular function \(\phi(s)\), I'd bet money that the only one that works is unique (I know it's unique because Kneser is unique), and so there's some modular function that makes Kneser.
This would mean that \(\beta\) would have a flavor of modular functions; it would still admit singularities but the singularities would look like discontinuities between \(\beta(s+1) = e^{\beta(s)}\) or \(\beta(s+1) = 0\), based on how \(e^{i\phi(s)s} = 0,\infty\) when \(\phi\) has a singularity. Additionally you'd be able to talk fluidly about the transformation \(1/\beta(s)\) is equivalent to \(z \mapsto -1/z\) much more fluidly (again, this was a large part of the paper for describing the weak Julia set).
This would mean we wouldn't get "singularities" in \(\beta\), we would get essential singularities that look like \(\beta(s+1) = 0,\beta(s+1) = e^{\beta(s)}\). These, in the end would become our values of kneser that equal the orbit \(\exp^{n}(0)\). This shit is unbelievably fascinating. But I have so much ODE/PDE research I've put on hold for 2 years to look at tetration; I need a break, rofl.
So, I'm going back into my dark little corner of studying Schrodinger's equation, and how it relates to infinite compositions. See you guys on the otherside
!!!Regards, James
