Yesterday we were treated to a public lecture by the enigmatic Marcus du Sautoy
on the music of the primes
. It began with a discussion of the shirt numbers in European football teams, something that du Sautoy was particularly enthusiastic about. Fortunately, by the end of the hour we were well into the Riemann hypothesis
He gave a wonderful explanation of what the zeroes mean. According to the hypothesis, the non-trivial zeros of the zeta function all lie on the critical line
, the real part of $z$ equal to 1/2. Instead of plotting the axes of the complex plane, however, he showed the line on a plane marked by a vertical axis for frequency
(of a Riemann harmonic component of the full prime counting function) and a horizontal axis for amplitude
So there is a way to label the zeroes one by one, starting with the one lowest on the positive vertical axis at $y$ = 14.1347 (and remembering that there is a reflection symmetry across the horizontal axis). The label counts the number of nodes in a wave of some kind. All waves have the same
amplitude. This fact must relate to the localisation we find with T-duality in twistor ribbon models. So a basic principle of quantum gravity forces the Riemann hypothesis to be true. What is this wave? The real world manifestation that du Sautoy mentioned was that of the energy levels for large atoms, such as uranium. But as we know, most analyses of the match between energy levels and Riemann zeroes involve the statistics of an ensemble. We would rather construct
the zeroes one by one. The expectation is that the algorithm, if it exists at all, would become increasingly complex, so that the computation of all
the primes would amount to a vast universal computation. But hang on a minute! Isn't that what our rigorous QFT/QG program is all about?
We learn in kindergarten that the energy levels of atoms correspond to the allowed shells for electrons. Thus there is a correspondence between $E_n$ and particle number. Now recall, we saw that the tricategorical aspects of QCD suggest a correspondence between particle number and categorical dimension. This is good, since we expect categorical dimension to provide a measure of complexity for a problem naturally couched in that context, such as calculating the nth zero.
Another clear aspect of du Sautoy's talk was his explanation of the logarithm
function, as a method for converting multiplication
. The logarithm acts as the first mode of Riemann's wave decomposition. We have met the higher dimensional analogues before in operad combinatorics.
Finally, there were a few fun film clips from movies about mathematics. As Robert Redford
says when the detective slots the codebreaker box into the aircraft control computer, No More Secrets