Did you read it here? Riemann Zeta Zeros
If not, then you should check it out. It's pretty interesting. Your surest bet is to read up on Random Matrix Theories and quantum chaotic systems, as the author mentions these.
I have read that statistical analysis of the distribution of the known non-trivial zeros of the Riemann zeta function has given a fractal dimension of about 1.9. Nice, but is there anything in any of the (non-statistical) equations that make the self-similarity evident?
Thank you, Vlasev. Yes, the site you mentioned was one of the places I read about the statistical take on the zeros. I am also roughly aware of quantum chaos , but pursuing that in connection with the fractal dimension would be taking me down the road to better understand the links between fractal structures and physics, rather than finding out whether the fractal dimension of 1.9 mentioned here is purely an empirical result (as it is in that article), or whether there is something in one of the equations used for the Riemann zeros to indicate a self-similarity. The field of Random Matrices is more promising, but there might be a more direct connection. For example, the fact that the functional equation in which the zeta function of s = an expression with the zeta function of (1-s) as a factor -- is this sufficient to indicate a self-similarity that results in a the distribution having a fractal dimension?
I think this paper is fascinating and shows how the zeta function is a fractal.