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.