I'm by no means an expert in Farey Sequences nor understanding the Riemann Hypothesis - but I do know there is some relationship between them, as alluded to in this question. As such, if the Mandelbrot Set with a -1 power (see image below) produces a Farey Sequence along the imaginary axis, could this be used to help solve/prove the non-trivial zeros for the Riemann Hypothesis? Has something like this already been attempted?
Note that the radii of the following circles will be reduced if either the bailout/threshold or the maximum iterations are increased, I don't know if they converge or not. The Escape-Time algorithm is used, so more detail might be revealed with a different algorithm.
Edit: On the subject, I noticed that as the Mandelbrot's power approaches zero, what would correspond to the first elements to the Farey Sequence move slower to the left:
I'm pretty sure that this is the Farey Sequence at this point, but all I have to show for it is just by looking at it. I wonder if algebraically solving for the boundary of this fractal will yield interesting insight...
Here is pixels $c$ coloured by iteration index $p$ where $|z_p|$ is minimized under iterations of $z_{n+1} = \frac{1}{z_n} + c$. The right hand side is clamped to $\Re(c) = 0.01$ to make the structure clearer. The Farey sequence seems to be visible (but this is no mathematical proof).