(02/08/2023, 08:40 PM)Caleb Wrote:(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
AHHH! I see, I apologize. I was confused!
Then yes we are on the exact same page
I've made some graphs I'll post in a bit, but the zeta series appears to converge for \(0 < \Re(s) \), where I presume we can analytically continue using Gottfried's expansion. I apologize!
If you are saying that:
\[
\sum (-1)^n n^{n^{-s}}
\]
converges for \(\Re(s) > -1\) we are on the same page. I actually haven't looked at this sum, I just instantly wrote it as a zeta function and only looked at that--so I wasn't sure where the original series converged.
It also appears the zeroes are on \(\Re(s) = 1/2\) on the critical strip... having Riemann Hypothesis vietnam flashbacks
So I accidentally closed one of the graphs mid compile, but the other one finished. So here is:
\[
\nu(s) = \sum_{m=1}^\infty q(m) m^{-s}\\
\]
Where:
\[
q(m) = (-1)^m \sum_{n \le m} \chi_m(n)\\
\]
And \(\chi_m(n) =1\) if and only if \(n^k = m\) for some \(k \in \mathbb{N}\).
Then here is a graph from:
\(0.5 \le \Re(s) \le 5.5\) and \(|\Im(s)| \le 2.5\):
Didn't finish the graph for gottfried's function, it only finished most of the top half, so I'm going to recompile. I am also graphing the critical strip for both, to see what that could look like.
