Are solutions of $\zeta(s) = 1$ very near a line $\Re(s) = 54$ and solutions of $\zeta(\zeta(s)) = 1$ either on or very near a circle with center $\approx .00936$ tangent to $\Re(s) = 1$, known to exist? I seem to be observing these things in computer experiments. The solutions to the first equation (zeta images of the solutions to the second one) look like they lie on circles of radius $\approx 4 \times 10^{-10}$ centered on points of the form $54 + i k \pi/\log(2), k$ odd.

Is this ridiculous? What might produce such an illusion?

Edit: To illustrate, here is a link to a PDF with graphics of the phenomenon: http://barrybrent.9f.com/zeta=1.pdf. (~ a meg.)

There are other such apparent circles for higher zeta iterates, which I'll show in a later draft, if the observations aren't knocked down.

Barry Brent