So I believe I've managed to prove that the minima/maxima cluster towards the real line; giving arbitrarily close singularities to the real line as the real argument of \( \phi \) grows. And, additionally, that solutions of \( \phi(s) + s = 0 \) necessarily cluster towards the real line; and cannot happen in the strip \( \delta < \Im(s) < 2\pi - \delta \) for large enough \( \Re(s) > T \) (here \( \delta \) depends on \( T \), and as \( T\to\infty \) we get \( \delta \to 0 \)).
Luckily I so worded the main results of this paper that this was never not a possibility. But I had implicitly assumed it would be analytic on \( \mathbb{R} \). I'm in the process of cleaning this up; and trying to make the argument a bit more solid than I have it right now. But I believe that this tetration should look like this,
\( e \uparrow \uparrow s : \mathbb{C}/(\mathbb{R} + 2 \pi i k) \to \mathbb{C} \) is analytic
\( e \uparrow \uparrow t + 2\pi i k \) is at least continuously differentiable--definitely probably nowhere analytic (but I can't prove it's nowhere analytic--can't really say much).
This of course still meshes with the statement \( e \uparrow \uparrow s \) is holomorphic on \( \mathbb{C}/\mathcal{L} \) for \( \mathcal{L} \) a nowhere dense set. It just looks a lot different than I was expecting.
Luckily I so worded the main results of this paper that this was never not a possibility. But I had implicitly assumed it would be analytic on \( \mathbb{R} \). I'm in the process of cleaning this up; and trying to make the argument a bit more solid than I have it right now. But I believe that this tetration should look like this,
\( e \uparrow \uparrow s : \mathbb{C}/(\mathbb{R} + 2 \pi i k) \to \mathbb{C} \) is analytic
\( e \uparrow \uparrow t + 2\pi i k \) is at least continuously differentiable--definitely probably nowhere analytic (but I can't prove it's nowhere analytic--can't really say much).
This of course still meshes with the statement \( e \uparrow \uparrow s \) is holomorphic on \( \mathbb{C}/\mathcal{L} \) for \( \mathcal{L} \) a nowhere dense set. It just looks a lot different than I was expecting.