I've been reading about integral transforms (Mellin transform and Fourier transform) and was wondering if there was a promising way to map the prime numbers into a different region? What if you could squash all the primes into a bounded, compact space, which might be easier to manage than having them all run off to infinity, and become more sparse as they go.
Edit 3/1/2020:
I think I'm looking for the notion of a continuous compactification of a meromorphic function. Specifically, is there a way to "track," so to speak the location of values of a meromorphic function undergoing "deformation." I'm thinking of something like a homotopy, but is there anything like that in complex analysis, or just in topology?
I mention primes because there is a well-known function that encodes rich information about the primes. Maybe if such a "complex homotopic map" was found, then progress could be made solving the Riemann Hypothesis. After a quick online search, I think this notion already exists! (https://www.fmf.uni-lj.si/~forstneric/papers/2003%20Contemp.%20Math.pdf)
Extending domain of a meromorphic function from non-compact to compact domain
Can anyone summarize (in a few sentences) the paper to I linked in parentheses to help me (and possibly others) understand the basics of such a homotopic map in the field of complex analysis?
Here's my attempt: "The homotopy principle for meromorphic functions is essentially a way to smoothly encode and transition geometrical and/or analytical information about a meromorphic function, assuming that there are no topological obstructions."