Quote:The ultimate state of what you are doing is analytic continuation, but it's the analytic continuation of sums. There's a reason every analytic continuation of sums uses the Mellin transform. All's I'm saying.No! This isn't whats happening in the post! My pedagogy in the post is terrible, so I apologize for not drawing attention to a crucial fact of whats happening in the post. I'm looking at summations past natural boundaries! By natural boundary I mean there are a dense set of singularites. This means no argument can be made on the basis of analytical continuation-- there is no analytical continuation for these series! This fact explains why the residues are not picked up for some domain, but then all the sudden picked up in another domain-- this automatically forces the function to be discontinuous at the point we decide to pick up the residues. Indeed, at the point where the extra residues get picked up is where the natural boundary is located. This is why I fuss about the canonical extension in the post-- becuase there isn't only one extension of a function beyond its natural boundary. However, I have chosen my series in my post very carefully-- I picked them so that they actually converge on both sides of the natural boundary, so that there is a natural extension of the function beyond the natural boundary. Actually, I think this is kind of a natural extension of the little circle method you had mentioned a few days ago. The little circle method takes arcs inside the circle to try to compute the residues. My approach is, essentially, to take a contour outside the circle.
There's also a second issue with the mellin transform approach, which is that it only works under certain growth conditions, and I don't limit myself to those growth conditions here. In general, my testing in the past suggests to me that when we start to consider functions with larger growth conditions, new behaviour starts to emerge that wasn't there with the slower growing function. Thus, we can't just straightforwardly extend the mellin transform approach to work in many of the cases I'm using the residue theorem in the post.
Also, I should add that ultimately, my goal in studying of all of this is to produce some theory about (non-analytic) continuation beyond natural boundaries. Ever since I saw the beautiful graphs of modular forms such as the Jacobi theta function two years ago, I've wondered what lies on the other side. Thus, these objects I'm studying are motivated by an early attempt to study the relationship between complex functions inside and outside their natural boundary so that I might eventually figure out the proper way to gaze upon modular forms in the lower half plane.
