I have a very vague intuitive idea which I have been struggling to give a meaningful definition to, and was wondering if this idea I am working on has been tried before. I apologize if I am unable to formulate these abstract intuitive idea's better than this; I am trying to work on my exposition for this type of question, so any advice is much appreciated.
I want to treat the analytic continuation of the zeta-function as some kind of conformal mapping between two regions, $A$ $B$ of the complex plane, with $\zeta : A \rightarrow B$, which satisfy the following two properties
- Neither $A$ nor $B$ are the entire complex plane
- One or both of $A$ or $B$ are not simply connected
One of the guiding principles I have is that the singularity at $z=1$ somehow represents "the point at infinity in the Riemann sphere" and to "treat the entire complex plane as a projective plane in the sense of projective geometry", and that the critical region between $z=0$ and $z=1$ is somehow related to "the topology of the projective plane"
Intuitively, at least for me, the projective plane is a "twisted up 2 dimensional space where you cover a watermelon with cellophane and then twist it off at the top and glue it back together in a non-realistic non-physical way"
More rigorously, we can represent the topology of the projective plane as a square, or so-called "fundamental rectangle", where the sides of the square are "glued together" in a specific way as shown in this picture
I just had the most strange idea which is to try and make the singularity of the zeta-function at $z=1$ a point at infinity in the projective plane, and to try and map the critical strip to the so-called "twist off" section of the projective plane which results from "gluing together" the fundamental rectangle in the described way
There are many reasons I was led to this kind of approach, but at this time I have no idea if anyone has ever tried this before, or if this is a very bad/silly idea that won't work. I will only provide one more piece of information about the motivation at this time, feel free to ask if you are curious in the comments.
The basic idea is to figure out if it's possible to show that the critical line, $Im(z)=\frac{1}{2}$, corresponds somehow to the "line at infinity" in the projective plane
(EDIT: Here is one way to view the complex plane as a projective plane: consider a 3-dimensional real space and imagine a unit sphere centered at the origin. Construct the projective plane using the standard definition of the set of all lines passing through the origin. Project all points of 3-d space onto the surface of this sphere with this construction in a 2-1 way; then project all the points on the surface of this sphere (except for one) onto the complex plane using the Riemann sphere projection.)