Self similarity of z/log(z)

I made the following pics by temporally iterating 3 spatial iterations;

z/log(z)
(z/log(z)) / log(z/log(z))
[(z/log(z)) / log(z/log(z))] / log[(z/log(z)) / log(z/log(z))]

The temporal iteration number is 170. Bailout is 40. The 4th pic is a close-up of the 3rd. They came about after a Google group discussion.

The 3 pictures similarity is amazing I thought. Just like the temporal iteration increasing resolution, the spatial iteration seems to do the same. I think there is a strong link here to the prime numbers, whose distribution is known to be self similar. Apart from the known fact that the primes are semi uniformly distributed in x/log(x), they seem to obey the H-fractal in their further patterns.

Another very interesting point here is that the both temporal and spatial iterations limit to 'e' at the outer region of the shape;

...(z/log(z)) / log(z/log(z))... -> e

demonstrating the binary structure of 'e'.

The tip of the last amplified to the limit of my pc. Note the slight angle. Numbers seem to prefer a side, as far as this function is concerned. This may be due to the accumulation of errors. If not, then this is also very interesting.

The semi-stable points. What is the link to the zeros of the Riemann's Zeta function?