Gluing leaves in zeta space and automorphisms on the resulting quotient spaces

Define an analytic planar homotopy $h_t: (0,1) \to (0,1)$ for $t\in[1/3,3]$ where $h_t(x)=e^{\frac{t}{\log x}}$ s.t. $h_{1/3}=e^{\frac{1/3}{\log x}}$ and $h_3=e^{\frac{3}{\log x}}$ for $x\in(0,1)$. I want a rotational homotopy that projects onto the planar homotopy. So, define the equivalence relation on the boundary curves of the closed subset of $(0,1)^2$, that is $h_{1/3}\sim h_3$, where points are identified by straight lines with slope $1$, connecting $h_{1/3}$ and $h_3$. This yields a spherical topology $\mathcal S$ embedded in $(0,1)^3$ with the endpoints of the sphere being singularities at $p=(0,1,1)$ and $q=(1,0,0)$. 

Conceptually it's clear that a projection $\pi: (0,1)^3 \to (0,1)^2$ takes the leaves on $\mathcal S$ to $h_t(x)$ as a double cover. But my main objective of study is to put a metric on $\mathcal S$. The $1$-parameter metric I derived on the planar space is $g_t(r)=\int_{(0,1)} x^{r-1}h_t(x)~dx = 2\sqrt{\frac{t}{r}}K_1(2\sqrt{tr})$ and it measures the distance between the $h_t(x)$ leaves. For example the distance between leaf $t=1/3$ and leaf $t=3$ is $d(1/3,3)= \bigg| g_{1/3}(r)-g_{3}(r) \bigg |.$

If one repeats this gluing process by taking $t\in[1/k,k]$ for every real $k>1$ then one obtains shells, at least mutually diffeomorphic to $\mathcal S$ embedded in $[0,1]^3$ which can be made into a smooth codimension one foliation of $(0,1)^3$ with $k=1$ corresponding to the only codimension two leaf. Call the codimension one set of shells, $\mathcal G$. 

Since the analytic homotopy $h_t(x)$ satisfies

$$t \frac{\partial^2}{\partial t^2} h_t(x) = - x \frac{\partial}{\partial x} h_t(x)$$

I would expect a compatible differential equation for $\mathcal G$ s.t. each leaf in $\mathcal G$ corresponds to an instant in time, as was such for $h_t(x)$. This is desirable as it would give a geometric flow that shrinks the surfaces asymptotically to the codimension 2 leaf. 

Some evidence for this already is that the metric $g_t(r)$ satisfies a linear third order PDE and geometric flow

$$t^2 \frac{\partial ^3}{\partial t^3}g_t(r) =r^2 \frac{\partial}{\partial r}g_t(r)$$

where we took the Mellin transform of the leaves of $h_t(x)$ to obtain the desired PDE.

Evidently, the leaves in $\mathcal G$ are compactly supported with sufficient decay properties giving access to the Paley-Wiener theorem, allowing one to conclude analyticity of the Fourier transform of these leaves. 

Now if we act on $\mathcal G$ by rigid automorphisms in $\Bbb R^3$ then the only allowable action is a $360$ degree rotation which leaves the set of leaves in $\mathcal G$ invariant. We can layer on additional symmetry by letting $\mathcal G$ be just one element and taking $\mathcal G_{\alpha}$ for $\alpha=1,2,3,4$ gaining octahedral symmetry. It can be shown that the Euler characteristic vanishes i.e. $\chi(M)=0$ where $M$ is just the union of the $\mathcal G_{\alpha}$ but that is another story. 

How do the automorphisms of $\mathcal G $ interact with the complex space? We know by the Paley-Weiner theorem that analyticity is inherited, but does an automorphism of $\mathcal G$ necessarily correspond to some invariance of the Fourier transform of a generic leave in $\mathcal G$? What I am looking for is some symmetric functional equation in $\Bbb C$ that is unchanged under an automorphism of a generic leaf in $\mathcal G$.

For example clearly $h_t(x)$ is involutive. It also satisfies $h(x^y)=h(x)^{1/y}$ which can be viewed as a eigenfunction problem. I am also reminded of the Mellin transform of the Jacobi third theta function completing the Riemann zeta function and through Poisson summation giving rise to a functional equation on the complex space. But I don't think Poisson summation applies here because $h_t(x)$ is not a series of functions.

Extending the base metric to the quotient metric in $\zeta$-space

Define an analytic planar homotopy $h_t: (0,1) \to (0,1)$ for $t\in[1/3,3]$ where $h_t(x)=e^{\frac{t}{\log x}}$ s.t. $h_{1/3}=e^{\frac{1/3}{\log x}}$ and $h_3=e^{\frac{3}{\log x}}.$ I want a rotational homotopy that projects onto the planar homotopy. So, define the equivalence relation on the boundary curves of the closed subset of $(0,1)^2$, that is $h_{1/3}\sim h_3$. This yields a spherical topology $\mathcal S$ embedded in $(0,1)^3$ with the endpoints of the sphere being singularities at $p=(0,1,1)$ and $q=(1,0,0)$. 

Conceptually it's clear that a projection $\pi: (0,1)^3 \to (0,1)^2$ can take the leaves on $\mathcal S$ to $h_t(x)$ as a double cover. I would also like to put a metric on $\mathcal S$. The metric I derived on the planar space is $g_t(r)=\int_{(0,1)} x^{r-1}h_t(x)~dx = 2\sqrt{\frac{t}{r}}K_1(2\sqrt{tr})$ and it measures the distance between the $h_t(x)$ leaves. For example the distance between leaf $t=1/3$ and leaf $t=3$ would be a subtraction of integrals $d(1/3,3)= \bigg| g_{1/3}(r)-g_{3}(r) \bigg |.$ This distance changes w.r.t. to $r$ as well because it's defined to be a $1$-parameter family of metrics. 

Some questions: How do we prove that after gluing with the equivalence relation, that $g_t(r)$ has an extension to $\mathcal S$? 

Comments: It's desirable to control the geometry under the action of the quotient. We'd like to be able to put the quotient metric on $\mathcal S$. 

Zeta space as a probabilistic model

Consider $ S^n,$ the space of Schur-convex, simply connected, closed topological $n-$manifolds as subsets of the unit $(n+1)-$cube, which include $p=(0,0,\cdot\cdot\cdot,0)$ and $q=(1,1,\cdot\cdot\cdot,1).$ Consider the set of all Schur-convexity preserving maps from $S^n$ to itself.

Define a class of $S^n,$ denoted $\zeta(\mathbf X),$ in the following way:

Let $ L^n_+$ be the set of all $n$-dimensional nonnegative random vectors $\mathbf X = (X_1, X_2,\cdot\cdot\cdot,X_n)^⊤$ with finite and positive marginal expectations, and let $\mathbf Ψ^{(n)}$ be the class of all measurable functions from $\Bbb R^n_+$ to $[0, 1].$ Then $\zeta(\mathbf X)$ of the random vector $\mathbf X$ with joint CDF $F$ is:

$$\zeta(\mathbf X)=\bigg\{\bigg(\int \psi(\mathbf x)dF(\mathbf x), \int \frac{x_1\psi(\mathbf x)}{E(X_1)}dF(\mathbf x),\cdot\cdot\cdot,\int \frac{x_n\psi(\mathbf x)}{E(X_n)}dF(\mathbf x):\psi \in \mathbf Ψ^{(n)}\bigg)\bigg\}, $$

$$ =\bigg\{\bigg(E\psi(\mathbf X), \frac{E(X_1\psi(\mathbf X))}{E(X_1)},\cdot\cdot\cdot,\frac{E(X_n\psi(\mathbf X))}{E(X_n)}:\psi \in \mathbf Ψ^{(n)}\bigg)\bigg\}. $$

Let me provide a concrete example. I have a 3D graph of $\zeta$ for a bivariate II Pareto distribution with parameters $(\mu_1,\mu_2,\sigma_1,\sigma_2)=(0,0,1,1)$ and $\alpha=9$ so I know the qualitative nature of the shape of any given surface given by any distribution. Here is a picture to give some intuition about the definition:

I was interested in obtaining the volume for $\zeta(\mathbf X).$ I calculated it to be:

$$\mathrm{Vol}(\zeta(\mathbf X))= \frac{1}{(n+1)!}E\big(|\det(Q)|\big). $$

where $Q$ is a $(n+1)\times(n+1)$ matrix whose $i$th row is $(1, \mathbf{ \bar X}^{(i)}),$ $i=1,2,\cdot\cdot\cdot , n+1.$

Here I'll use a normalized version of $\mathbf X$ denoted by $\mathbf{\bar X}$ such that $\bar X_i= X_i/E(X_i),$ $i=1,2,\cdot\cdot\cdot, n.$ Consider $n+1$ $\mathrm{iid}$ $n$-dim. random vectors $\mathbf{ \bar X}^{(1)},\cdot\cdot\cdot, \mathbf{ \bar X}^{(n+1)} $ each with the same distribution as $\mathbf{\bar X}.$ 

I was interested in the Mellin transform of the volume form of $\zeta$ which can be written as:

$$ \Pi_n(s)=\int_{\zeta} x^{s} ~\mathrm{Vol}(\zeta(\mathbf X)) \frac{dx}{x} $$

The reason for investigating this object is to construct a symmetrical completion of $\zeta(\mathbf X)$ in the sense that $\zeta(\mathbf X)$ is just one component in the space. We can add three more of these components to obtain $\zeta_{\mathrm{complete}}.$

Then one can take Mellin transforms of the volume forms on each of the components, paying particular attention to how the mutual intersection volume region transforms into the image space under the Mellin transform. Call this region $R$ which is a subset of $\Bbb C$.

> What can be said about $R$?

Automorphisms and zeta space

Here I will operate under the assumption that $\phi_s$ is a unique and well-posed solution to the following PDE. This means whatever conditions are needed for that to occur, will be imposed.

$$s\frac{\partial^2}{\partial s^2}\phi_s(x)=\mp x\frac{\partial}{\partial x}\phi_s(x) $$

for which we have our pre-assigned unique solution:

$$\mathcal \phi_{s}:=\bigg\lbrace e^{\frac{\pm s}{\log x}}:s \in \Bbb R \bigg\rbrace$$

We have singularities at $s=0$ and $x=0,1$ and this will be called a "cross singularity" or  $CS$. A natural idea would be to intialize some distribution on this $CS$ and run time forward giving a wellposed analytic solution for all time. So we choose our pre-assigned solution $\phi_s$ and run the solution back in time where we obtain what we wanted, at least qualitatively.

We can throw out our solution for $s<0$ because it doesn't directly apply to the situation, and it does not rapidly decay. Meanwhile, for $s>0$ it is Schwartz i.e. has a rapid decay criterion, and we know its Mellin transform exists and has a closed Bessel form. This will be employed to extract a *1-parameter family* of functional equations for the Riemann zeta function.

$$\Gamma(\phi,r,s)\cdot \zeta(s) = \Gamma(\hat{\phi},r,1-s)\cdot \zeta(1-s)$$

whereby moving $\zeta(s)$ to the RHS gives us a partial differential relation on the factor $\Gamma(\phi_s,r)$

$$r^2 \frac{\partial ^3}{\partial r^3}\Gamma(\phi_s,r) =s^2 \frac{\partial}{\partial s}\Gamma(\phi_s,r) $$

where we recognize that the Mellin transform of our Schwartz family, $\phi_s$ must obey a third order (shallow water wave) PDE. Here we explicitly calculate the factor which we can quickly verify as a solution.

$$\Gamma(\phi,r,s)=\int_{\mathbb R^\times \cap ~(0,1)} |x|^r~\phi_s(x)~{dx\over |x|}=2 \sqrt{\frac{s}{r}}K_1(2\sqrt{r s})$$

My motivation at least, stems from building a space, a so called zeta-space, or $\zeta$-space which somehow encodes all of what I wrote above and goes further, letting one express automorphisms of $\zeta$-space as leaving critical strips invariant under these automorphisms. 

I will now transition to more visual diagrams in part because it's much easier to convey the point and because I am much less sure how to rigorously show the mathematics in this part of the post.

$\phi_s$ is now shown to be decorated on all the panels of the unit cube in $\Bbb R^3$. I left out two panels because it would get cluttered. Here $\phi_s$ are understood to be our $1$-parameter family of Schwartz functions (the solutions we did not discard from the original 2nd order PDE).

We can build sections within the cube by gluing bounded subsets of the $\phi_s$ solutions to the original 2nd order PDE and requiring (by definition of a section) that projections take us onto the leaves, $\phi_s$ on the base space. If we do this correctly, the sections should look like this:

Finally we symmetrize to get symmetrical $\zeta$-space:

This is not rigorously posed yet because I haven't figured out how to say it precisely, but if the 3 rubber band loops around the object allow the object to act like a Rubiks cube group we can get a permutation structure on zeta-space. Any Rubiks cube transformation leaves the boundary invariant i.e. leaves the Schwartz families unchanged corresponding to an invariant quantity of the automorphisms of zeta-space.

Representation of Zeta space from a PDE and combinatorial viewpoint

Consider the following partial differential equation:

$$ s \frac{\partial^2}{\partial s^2} \phi(x,s) = - x \frac{\partial}{\partial x} \phi(x,s) $$ 

with our classical analytic generator as in the previous post, $\phi_s(x)=e^{\frac{s}{\log x}}$. This local foliation generates zeta space on the base planar space. 

Now, each leaf of the foliation directly corresponds to a Schwartz function therefore we can build out corresponding symmetric function equations and gather critical strips for each leaf.

The goal going forward will be to establish a very particular generalization of the leaves on the base space and allow them to sit in $\Bbb R^3$. They will form football like structures and the football structures can be seen as sections sitting over top of the base space. In fact, the base space of the foliation will be pasted onto each "face" of the corresponding cube, wherein the football structures will be promoted within the cube.

Then we will study the combinatorics of zeta space and relate the combinatorial information to the Riemann zeta function itself - most importantly giving a firm connection between the combinatorics and permutational information about zeta space, and that of the nontrivial zeros in the critical strip.

Now, we have pasted planar zeta space across the cube boundary - and here is an image of what that resembles:

This automatically defines a football structure within the cube defined by the boundary information.

But this is only one "component" of zeta space in the cube. We symmetrize allowing to decorate the boundary faces by a reflected copy of planar zeta space. Here is what the symmetrization of zeta space in the cube looks like:

Note that the combinatorics are very rich here. And the key insight is to understand each white strand to encode a deformation of a Riemann zeta function, precisely each white strand should be thought of as encoding a critical strip with nontrivial zeros. 

What happens when we think of the "rubber band curves" as rotational? Then this becomes a combinatorial gadget. And any twist of zeta space implies a scrambling of zeta space. 

So we must ask about the significance of cutting along say the blue rubber band curve, turning the lower half of zeta space by one unit and re-gluing. This really says that we are slicing the critical strip in certain places, and performing surgery, then a transformation on the pieces, then gluing things back in a way that looks like nothing has been changed at all. This is a symmetry, or an automorphism of zeta space.

Now, remember that we can view these transformations from the boundary point of view only. We will see an incomplete picture here though as the boundary is completely preserved! It's invariant. That means that if we only can see the boundary information we see no permutation or scrambling occurring. Perhaps we decide to color different zeta space on the cube faces and then we can see colors changing but still there is a nice invariance here.

What does that even tell us though? You might be thinking that there's no link between the combinatorics and the Riemann zeta nontrivial zeros.

But there is. If the critical lines are invariant under automorphisms of zeta space, then they are preserved quantities - we can use this thinking to represent the critical line and zeros as qualitative features on zeta space.

It becomes very apparent that if we want to prove that all the nontrivial zeros lie on the real line $\sigma=1/2+it$ then we need to simply understand the automorphisms of zeta space.