02/08/2023, 08:40 PM
(02/08/2023, 05:52 AM)JmsNxn Wrote: I think our mix up is you are using Gottfried's function. I am changing the variables \(s \mapsto -s\)--and then I am finding that this is actually a zeta function. And then I am wondering if we can analytically continue the zeta function. Just because it "looks" like a natural boundary, does not mean it is one. But that'd be super cool if it is. That would mean we have TWO distinct functions, one holomorphic for \(\Re(s) > 1\) and one holomorphic for \(\Re(s) < -1\); and there's a bunch of mystery there. Despite both "formally" equaling Gottfried's series.
Are you proposing that the change of variables is actually doing something to effect the convergence? I had not been making an distinction about changing variables, since I was under the impression that to go from Gottfrieds function to your function is a simple matter of replacing s with -s. Is Gottfried's function not simply a zeta function in the variable -s? I'm aware its possible that sometimes formally changing a variable can end up leading to a different function (for instance, maybe it ends up on a different branch), is that what your saying is happening here? Also, the extension I have for Gottfrieds function \( \sum (-1)^n n^{n^x} \) goes up to \( \Re(s) < 1 \) not \( \Re(s) < -1 \). This should correspond to \( \Re(s) > -1 \) for your function. I suspect you should find much more choas as you get towards that line
