"I...appreciate your sending me a copy of the Castro et. al. paper. I had not seen it before. However, now that I have taken a look at it, I find it hard to take it seriously. They don't write down explicitly where they think a counterexample to the Riemann Hypothesis might lie. In general, their whole approach is wrong-headed. One does not write a scientific paper reporting on a possible bug. One investigates the bug first, which they have not done. One can use programs other than Mathematica, or even write code from scratch. There are obvious problems with the results that they do report. For example, in the first table (on page 9), they present values of dim(E_(s)), where the real part of E_(s) is large and positive. For such values, zeta(dim(E_(s))) is clearly 1 (just consider the ordinary Dirichlet series defining zeta), so all the values in the last column, both the large ones as well as the zero ones, are clearly spurious."

"Thank for sending us the Andrew Odlyzko's comments. I must to confess that he is right: The title of the article and the importance given to the bugs were wrong. As you say the relevant point of the paper is the relations of the zeta function with the fractal p-branes in Cantorian-fractal space-time, which was lost within irrelevant considerations about the spurious zeroes.

Also we learned that the zeta can not have zeroes at Re[s]<0, and Odlyzko teach us that zeta for very large Re[s] is one (actually all the arguments of zeta given at page 9 are all large and NEGATIVE, not positive as Odlyzko says).

The real important point of the paper lies in section 2.1 where was presented an heuristic "derivation" of the conjecture. It all rest on the existence of those fractal/discrete/p-Adic derivative operators with the adequate self-adjoint properties.

Quantum Groups emerged as a result of inverse scattering methods. The search is to find now whether such operators can be constructed. If they can, then their spectrum will pick up the imaginary part of the zeroes of the zeta.

About fractional spin and problems with the spin statistics theorems: The spin we found is just 1/2 the dimension of the fractal path of a boson in Cantorian-Fractal space-time = 1 + Golden Mean.

In conformal field theory the central charge of a boson is 1. That of a fermion is 1/2.

The same argument applies here but now we that the fractal path has dimension 1 + Golden mean for bosons and 1/2 (1 + Golden Mean) for fermions.

We need to examine the validity of the Riemann-Roch theorem for fractal Riemann surfaces, etc...

It is all heuristic but it may lead to a key to proving the Riemann conjecture.

About the Golden Mean Bug: We were misled by Mathematica."