Brainlet here, but it seems like mathematicians would be a lot more successful solving this if they didn't focus so much on integers. Integers are artificial constructs preventing a more fundamental understanding of whats going on here. So rather than just summing integers like
(1^-s)+(2^-s)+(3^-s)
they need a mathematical tool that takes all relative contribution between integers into account. Mathematicians act like pi showing up in the equation is some deep fundamental truth but it actual seems more like an artifact of integers to me.