02/16/2023, 09:51 AM
Alright; so a quick correction. It appears we lose steam as we go deeper into Gottfried's zeta functions. So that:
\[
\zeta^{(2)}_G(s) = \sum_{n=1}^\infty (-1)^{n+1} n^{n^{n^{-s}}}\\
\]
Only looks to be holomorphic for \(\Re(s) > 1\) (at least the zeta sum only converges here by the looks of it). My guess is that we are compounding too many functions, and our growth is no longer \(\log(x)\) but \(x \log^2(x)\). If I were to wager a guess; this continues for \(\zeta^{(K)}(s)\), where we get growth like \(x^{K-1}\log^{K}(x)\) in the coefficients.
If we write:
\[
h(n,k) = \sum_{j=0}^\infty \frac{\log(n)^j}{j!}j^k\\
\]
And:
\[
g^{(2)}(m) = (-1)^{m+1} \sum_{n\le m} h\left(n,\frac{\log(m)}{\log(n)}\right) \frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!} \chi_m(n)\\
\]
\[
A^{(2)}(x) = \sum_{m \le \lfloor x \rfloor} g^{(2)}(m)\\
\]
Then these seem to be growing much faster than \(\log(x)\); but they teeter out; and it looks close to \(x\log^2(x)\). At least the first 1000 terms seem to behave like this. We also see much more clearly a pole in \(\zeta_G^{(2)}(s)\) at \(s=1\); where for \(\zeta_G\) this happens at \(s=0\). So it seems pretty safe to say we are only holomorphic for \(\Re(s) > 1\). I think \(\zeta_G^{(K)}(s)\) is holomorphic for \(\Re(s) > K-1\) for \(K\ge 1\); and most of my trials seem to be supporting this.
Here is the function \(g^{(2)}(m)\) from \(1 \le m \le 1000\):
And a comparison graph of \(x\log(x)^2\):
Here is a graph of \(\zeta^{(2)}_G(s)\) for \(1 \le s \le 4\)--the pole is pretty obvious.
Here is the function \(A^{(2)}(x)\) from \(1 \le x \le 1000\):
[To be posted when it finishes compiling
]
Honestly, I'm surprised by how fast this chaos is balancing out. It's a real fucking headache to program; but I'm sure this is working to an extent! Jesus these are some fucked up zeta functions lmao.
Regards, James
\[
\zeta^{(2)}_G(s) = \sum_{n=1}^\infty (-1)^{n+1} n^{n^{n^{-s}}}\\
\]
Only looks to be holomorphic for \(\Re(s) > 1\) (at least the zeta sum only converges here by the looks of it). My guess is that we are compounding too many functions, and our growth is no longer \(\log(x)\) but \(x \log^2(x)\). If I were to wager a guess; this continues for \(\zeta^{(K)}(s)\), where we get growth like \(x^{K-1}\log^{K}(x)\) in the coefficients.
If we write:
\[
h(n,k) = \sum_{j=0}^\infty \frac{\log(n)^j}{j!}j^k\\
\]
And:
\[
g^{(2)}(m) = (-1)^{m+1} \sum_{n\le m} h\left(n,\frac{\log(m)}{\log(n)}\right) \frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!} \chi_m(n)\\
\]
\[
A^{(2)}(x) = \sum_{m \le \lfloor x \rfloor} g^{(2)}(m)\\
\]
Then these seem to be growing much faster than \(\log(x)\); but they teeter out; and it looks close to \(x\log^2(x)\). At least the first 1000 terms seem to behave like this. We also see much more clearly a pole in \(\zeta_G^{(2)}(s)\) at \(s=1\); where for \(\zeta_G\) this happens at \(s=0\). So it seems pretty safe to say we are only holomorphic for \(\Re(s) > 1\). I think \(\zeta_G^{(K)}(s)\) is holomorphic for \(\Re(s) > K-1\) for \(K\ge 1\); and most of my trials seem to be supporting this.
Here is the function \(g^{(2)}(m)\) from \(1 \le m \le 1000\):
And a comparison graph of \(x\log(x)^2\):
Here is a graph of \(\zeta^{(2)}_G(s)\) for \(1 \le s \le 4\)--the pole is pretty obvious.
Here is the function \(A^{(2)}(x)\) from \(1 \le x \le 1000\):
[To be posted when it finishes compiling
]Honestly, I'm surprised by how fast this chaos is balancing out. It's a real fucking headache to program; but I'm sure this is working to an extent! Jesus these are some fucked up zeta functions lmao.
Regards, James
