Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T =\Omega_T(\sigma)$ as the maximum circle, centered at $t_T$ on the real (horizontal) axis, for which $\eta(\sigma +it)\notin \Omega_T$ if $0< t\leq T$. In short, $\eta$'s orbit never enters the hole. Here $t_T=t_T(\sigma)$ is a function of $\sigma$, with $0\leq t_T < 2$.
If $\sigma>0.65$, the hole is very apparent for small values of $t$, say $t<10^4$. Or if you look at any bounded interval $t\in[T_1,T_2]$. It is part of the repulsion domain of $\eta(\sigma+it)$. Let $\rho_T=\rho_T(\sigma)$ be the radius of the hole.
My question: does the hole eventually vanish if $T$ is large enough, or do we have $\rho_T\rightarrow 0$ as $T\rightarrow\infty$ (yet $\rho_T>0$ for all $T$), or do we have $\lim\inf \rho_T > 0$, for some $\sigma<1$?
If $\sigma=\frac{1}{2}$ the hole is empty and stays empty as soon as $T$ hits the first non-trivial root of the Riemann zeta function. Actually it becomes the opposite of a hole: an attraction point of the orbit. But what about $\sigma=0.9$ or $\sigma=1.2$? Empirical evidence suggests that the hole moves to the left on the real axis, and shrinks, as $\sigma$ gets closer and closer to $\frac{1}{2}$. It the limiting hole ($T\rightarrow\infty$) encompasses the origin $(0,0)$ for a given $\sigma$ and is non-empty, it means that this particular value of $\sigma$ satisfies the Riemann Hypothesis.
Related to this question is whether the image of $\eta$ in the complex plane is bounded, especially bounded to the left. That is, can the real part of $\eta(\sigma+it)$ be arbitrarily negative for a fixed $\sigma$? The answer is clearly yes if $\sigma=\frac{1}{2}$ and clearly no if $\sigma=1.2$. But what about $\sigma=0.9$?
The goal is to study the attraction/repulsion domains of $\eta$ for a fixed $\sigma$, seen as a dynamical system, by looking at the orbit.
Update: see animated Gif with $\sigma=\frac{1}{2}$ (red), $\sigma=0.75$ (blue) and $\sigma=1.2$ (yellow). It shows points sampled on the three respective orbits. The hole for the blue orbit is clearly visible and encompasses $(0,0)$ The yellow orbit also has a hole. It would be interesting to know what the boundary of the yellow orbits is. The two others are unbounded, and the red one has no hole.
The picture is too large for this website, you can see it https://github.com/VincentGranville/Visualizations/blob/main/Source-Code/riemannFinalConfetti.gif
