Discussion on "tetra-eta-series" (2007) in MO

[Caleb]

I'm going to move on to the second function Gottfried was talking about. So let's write:

\[
T(-s,3) = \zeta_G^{(2)}(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{n^{n^{-s}}}\\
\]

This function is also holomorphic for \(\Re(s) > 0\). And I suspect for all higher orders. We start by writing:

\[
\zeta_G^{(2)}(s) = \sum_{k=0}^\infty \sum_{n=1}^\infty (-1)^{n-1} \frac{\log(n)^k}{k!} n^{kn^{-s}}\\
\]

We expand again, and get:

\[
\zeta_G^{(2)}(s) = \sum_{k=0}^\infty \sum_{j=0}^\infty \sum_{n=1}^\infty (-1)^{n-1} \frac{\log(n)^k}{k!} \frac{\log(n)^j}{j!} k^j n^{-js}\\
\]

Let's collect terms as a zeta sum:

\[
\zeta_G^{(2)}(s) = \sum_{m=1}^\infty g^{(2)}(m)m^{-s}\\
\]

We start by summing across \(k\) first; in which we get:

\[
h(n,j) = \sum_{k=0}^\infty \frac{\log(n)^k}{k!}k^j\\
\]

Where then we get:

\[
\zeta_G^{(2)}(s) = \sum_{j=0}^\infty \sum_{n=1}^\infty (-1)^{n-1} h(n,j)\frac{\log(n)^j}{j!} n^{-js}\\
\]

We are then given the formula:

\[
g^{(2)}(m) = (-1)^{m-1} \sum_{n\le m} h(n,\frac{\log(m)}{\log(n)})\frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!} \chi_m(n)\\
\]

And this function has the exact same growth as we saw before. Whereupon \(\zeta_G\) and \(\zeta_G^{(2)}(s)\) are at least holomorphic for \(\Re(s) > 0\).

I conjecture this continues for all \(\zeta_G^{(K)}(s)\) for all \(K\ge 0\).

---

\(K=3\) seems to behave this way too! Let's take:

\[
\zeta_G^{(3)}(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{n^{n^{n^{-s}}}}\\
\]

Then:

\[
\zeta_G^{(3)}(s) = \sum_{k=0}^\infty \sum_{j=0}^\infty \sum_{i=0}^\infty \sum_{n=1}^\infty (-1)^{n-1} \frac{\log(n)^k}{k!}\frac{\log(n)^j}{j!} \frac{\log(n)^i}{i!}k^j j^i n^{-s}\\
\]

Sum \(k,j\) first:

\[
h^{(2)}(n,i) = \sum_{k=0}^\infty \sum_{j=0}^\infty \frac{\log(n)^k}{k!}\frac{\log(n)^j}{j!}k^j j^i\\
\]

Which is simply:

\[
h^{(2)}(n,i) = \sum_{j=0}^\infty h(n,j) \frac{\log(n)^j}{j!}j^i\\
\]

Whereupon:

\[
g^{(3)}(m) = (-1)^{m-1} \sum_{n\le m} h^{(2)}(n,\frac{\log(m)}{\log(n)}) \frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!} \chi_m(n)\\
\]

Where:

\[
\zeta_G^{(3)}(s) = \sum_{m=1}^\infty g^{(3)}(m) m^{-s}\\
\]

The rule appears to be quite clear:

\[
h^{(K)}(n,k) = \sum_{j=0}^\infty h^{(K-1)}(n,j) \frac{\log(n)^j}{j!}j^k\\
\]

Where then:

\[
g^{(K)}(m) = (-1)^{m-1} \sum_{n\le m} h^{(K-1)}(n, \frac{\log(m)}{\log(n)}) \frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!} \chi_m(n)\\
\]

And:

\[
\zeta^{(K)}_G(s) = \sum_{m=1}^\infty g^{(K)}(m)m^{-s}\\
\]

Which are all holomorphic for \(\Re(s) > 0\). .....



I know it's not a proof yet; but I'm too lazy to write a whole paper to prove this. At most I'm going to do a notice, lmfao!    \(h^{(K)}\) is an iterated summation operator; which just looks like an integral operator. So we are turning an iterated number of integrals \(K\) into how deep \(\zeta_G^{(K)}\) goes. Gottfried would refer to these as linear operators; or linear systems of Taylor Series.

Call \(h^{(0)} = 1\); and:

\[
h^{(K)}(n,k) = \sum_{j=0}^\infty \frac{\log(n)^j}{j!} h^{(K-1)}(n,j)j^{k}\\
\]

Then we can rewrite this as an operator \(\mathcal{H}\) which takes functions taking \(\mathbb{N}^2 \to \mathbb{R}^+\). So that \(h : \mathbb{N}^2 \to \mathbb{R}^+\); and \(\mathcal{H} h : \mathbb{N}^2 \to \mathbb{R}^+\).  That operator is precisely:

\[
\left\{\mathcal{H} h \right\} (n,k) = \sum_{j=0}^\infty \frac{\log(n)^j}{j!}h(n,j)j^k\\
\]

Where we write all of Gottfried's zeta functions as:

\[
\begin{align}
\zeta_G^{(K)}(s) &= \sum_{m=1}^\infty g^{(K)}(m)m^{-s}\\
g^{(K)}(m) &= (-1)^{m-1} \sum_{n\le m} \left\{\mathcal{H}^{K-1} 1\right\}(n,\frac{\log(m)}{\log(n)}) \frac{\log(n)^{\frac{\log(m)}{\log(n)}}}{\frac{\log(m)}{\log(n)}!}\chi_m(n)\\
\end{align}
\]

Also; which is quick to grab; if \(h(n,k)\) is polynomial growth in \(\log n\) and \(k\); then \(\mathcal{H} h\) is too. And then, we are taking \(\log(m)\) growth within each polynomial. So we get polynomials in \(\log(m)\) growth. So there is no divergence, and \(m^{-\epsilon} g^{(K)}(m) \to 0\)--which looks like \(m^{-\epsilon} \log^K(m)\); and the oscillation between positives and negatives still happen. So we still have ABSOLUTE convergence of the zeta function series for \(\Re(s) > 1\), and CONDITIONAL convergence of the zeta function series for \(0 < \Re(s) \le 1\)...

This is for you matrix nerds  I might program this in my recursive language if you guys are interested. I can make some nice and efficient code...

I like the graphs, and I think this generalization into arbitrary heights is exciting! I'm really interested in the behaviour near the natural boundary, and in the region \(-1<\Re(s) < 0\), is it possible you could create some graphs in that region? I haven't really thought of a reasonable computational way to analytically continue the function there, do you have any thoughts?