Let $\zeta\colon\mathbb{C}\to\mathbb{C}$ denote the Riemann zeta function, analytically defined on $\mathbb{C}$ by meromorphic continuation, where $z \in D$. Similarly, let $G$ represent the co-domain of $z$ under $\zeta(z)$, such that $\zeta(z)\in G$ if $z \in D$. Is there a necessary condition that can be imposed on $z$, such that $\zeta(z)\in\mathbb{C^+}\cup \{0\}$ (i.e. $\zeta(z)≥0$ if $z \in F\subset D$) and $\zeta(z)\in\mathbb{C^-}$ (i.e. $\zeta(z)<0$ if $z \in H\subset D$) ?
Best Regards
This is approximately hopeless. Consider that every zero of the Riemann zeta function in the critical strip is (under the famous hypothesis) the intersection of a homeomorphic copy of the real axis and a copy of the imaginary axis, with the positive portion of the real axis extending to the right (and sometimes turning back to the left) and the negative real and both imaginary rays extending to the left.
Failure of the famous hypothesis would allow multiple copies of the axes (if the root was a multiple root) and axes to not extend off to the left or right (for a pair of roots not on the critical line). This is illustrated here (from "Riemann Zeta Function Zeros." From MathWorld--A Wolfram Web Resource)
The zeroes in the critical strip are the column of points in the right quarter. Points where zeta is pure real are blue. Points where zeta is pure imaginary are red. Each critical root has positive and negative imaginary axes running to the left. The negative real axis runs to the left. The positive real axis sometimes extends to the right to infinity and sometimes extends to the right a bit and turns around. The additional horizontal stripes of real values not attached to a root have not been much studied.
The sets you are interested in are various pieces of the blue curves in the image.