Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Friday, February 16, 2007

No More Secrets

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 into addition. 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.


Blogger CarlBrannen said...

Re movies. I recently found a wonderful site that has the scripts for many many famous movies. In addition to being slow on physics, I'm also slow watching movies. The scripts explain what the audience was supposed to intuit.

February 16, 2007 6:37 PM  
Blogger Mahndisa S. Rigmaiden said...

02 15 07

Hello Kea:
Now that duSautoy talk sounds exceedingly interesting!!!

Heheheh Now to briefly touch upon this comment:

"Another clear aspect of du Sautoy's talk was his explanation of the logarithm function, as a method for converting multiplication into addition. "

heheheh My dear Kea, this is what I have been going over quite a bit lately. I was fooling around with my p-adics the other night and trying to express 1/3 in base 2. Yeah, I know you can find this out anywhere on the web, however, it was important for me to derive the expansion myself for a few reasons.

One of them is that the richness of the equation a^x=p; where p is prime, and a is any real base.

The solution is found by expressing a^x as exp(log(a^x))=p, taking the log of both sides and seeing that:


From that, we can see that the Series Expansions for numbers in different bases are different and that the series is a way to read out how one might represent a number in an arbitrary base.

If x=log(p)/log(a) is an integer, then the a-representation of p is quite easy and will be represented by a 1 in an appropriate location. However, if x is rational, our expression for p may be easy, but expressing RECIPROCAL powers of p will be quite difficult and we will see non terminating decimals.

In base 10, 10/11=0.909090...In base 2, 1/3=0.01010101...
But even deeper, the series for x consists of additive terms but x=log(p)/log(a) so there is a DIRECT correlation between addition and the log function. And I forgot to mention that the series expansions for numbers expressed in base a all are geometric series, which we know SUM to known formuli.

Oh dear, I ramble. Thanks for such an interesting and stimulating post. I will be posting more thoughts on this issue in the next coupla days.

February 16, 2007 6:48 PM  
Blogger Kea said...

Hi Carl! Mahndisa, if you look at du Sautoy's publications he does quite a bit of work with p-adics - unfortunately mostly way over my head, but I'm going to take a look at the motivic zeta functions paper sometime. Motives are cool.

February 16, 2007 8:06 PM  
Blogger Mahndisa S. Rigmaiden said...

02 16 07

Hello Kea:
Thanks for the link! I make a correction to my previous post, p is not a prime, it is an arbitrary real number. If p is prime, we will most certainly NOT get an integer value for our x ratio IF we are doing a prime-adic expansion. If we are doing an a-adic expansion with a only real, then my previous statements are correct:)

February 16, 2007 10:30 PM  

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