The image of the imaginary-positive critical line $\frac{1}{2}+it, t \ge 0$ in the function $ዘ(z) = \frac{\zeta(z)}{\zeta^{\prime}(z)}$ (i.e. the reciprocal of the logarithmic derivative of $\zeta$) is a surprisingly neat spiral tangent to the imaginary axis at $0$ for values of $t$ corresponding to nontrivial zeros:
$\sigma={1\over 2}$" />This spiral is continuous and always winds counterclockwise.
For $0 < \sigma < {1\over 2}$ the spiral is less neat and self-intersects wildly, but it also seems to remain continuous (i.e. no loops seem to be pulled apart to infinity) and CCW-winding (i.e. no dimples or cusps appear in loops) ($\sigma=0.3$):
$\sigma={3\over 10}$" />Is it known to be true? If the truth of this conjecture ("loop conservation") is unknown, is it known to either imply or follow from RH?
EDIT: this "loop through $0$, tangent to the imaginary axis" behavior is not specific to $\zeta \over {\zeta^\prime}$. It is a universal property of $f \over {f^\prime}$ for any complex-differentiable $f$: the image in $f \over {f^\prime}$ of a small segment through a zero of $f$ is an arc through the origin, with the tangent at the origin parallel to the segment.
I was aware of this general fact when writing my question, but I was unaware of another fact that I only realized just now: this works for zeros of any multiplicity! $f \over {f^\prime}$ does not do what $f$ does, i.e. fly through $0$ at odd zeros and bounce from $0$ at even zeros. It always flies through $0$ at zeros of $f$. In fact, the (limit of) the derivative of $f \over {f^\prime}$ at a zero of $f$ is identically equal to the real number $1 \over m$, where $m$ is the multiplicity of the zero. This derivative being real makes sure that the arc at $0$ is parallel to the segment.
The purpose of this long edit is to point out that the basic "looping" behavior of $ዘ(\sigma + it)$ probably isn't "ruined" by non-simple zeros (at least not directly).