So I tried something a little whacky, I'll see if some interesting graphs spawn.
I'm trying to observe how:
\[
\beta_{y-e,1} = \Omega_{j=1}^\infty \frac{e^{(y-e)z}}{e^{j-s} + 1}\,\bullet z\\
\]
Behaves under iterated logarithms. This is the \(2 \pi i\) periodic function such that:
\[
e^{(y-e)\beta_{y-e,1}(s)}/(1+e^{-s}) = \beta_{y-e,1}(s+1)\\
\]
This object is holomorphic everywhere in \(s\) excepting singularities at \(j \pm (2k+1) \pi i\) for \(j \ge 1\) and \(k \in \mathbb{Z}\). and entire in \(y\)
I know this object converges under iterated logarithms for \(y \ge 0\), but I haven't really looked at what happens if we move \(y\) more to the left and leave Shell thron. If it has domains of normality near zero, it could mean that we may be able to massage a nontrivial solution in the neighborhood of \(b = \eta_-\). I'll see what happens.
I'm trying to observe how:
\[
\beta_{y-e,1} = \Omega_{j=1}^\infty \frac{e^{(y-e)z}}{e^{j-s} + 1}\,\bullet z\\
\]
Behaves under iterated logarithms. This is the \(2 \pi i\) periodic function such that:
\[
e^{(y-e)\beta_{y-e,1}(s)}/(1+e^{-s}) = \beta_{y-e,1}(s+1)\\
\]
This object is holomorphic everywhere in \(s\) excepting singularities at \(j \pm (2k+1) \pi i\) for \(j \ge 1\) and \(k \in \mathbb{Z}\). and entire in \(y\)
I know this object converges under iterated logarithms for \(y \ge 0\), but I haven't really looked at what happens if we move \(y\) more to the left and leave Shell thron. If it has domains of normality near zero, it could mean that we may be able to massage a nontrivial solution in the neighborhood of \(b = \eta_-\). I'll see what happens.
