M Theory Lesson 67

Terence Tao tells us about this paper by Guangming Pan and Wang Zhou on random $NxN$ complex matrices with entries of mean 0 and variance $\frac{1}{N}$. They claim to prove, under an assumption about moments, that the spectral distribution converges to the uniform distribution over the unit disc. This is called the circular law. In other words, the initial clustering of eigenvalues around the real line disappears as $N \rightarrow \infty$.

In would be interesting to see what this meant for honeycomb patterns in the limit of an infinite number of hexagons. Or perhaps it helps us understand the distribution of Farey numbers on the unit interval. Recall that successive terms $\frac{p}{q}$ and $\frac{r}{s}$ of a Farey sequence satisfy

$qr – ps = 1$

which is why the modular group appears when considering matrices $(r,p;s,q)$. Let $N$ be the number of terms in a Farey sequence. The Riemann hypothesis [1] is equivalent to the statement that the sum of differences between Farey terms and interval markers, namely

$\sum_{n=1}^{N} \delta_n \equiv | f_{n} – \frac{n}{N} |$

is bounded by $o (x^{\frac{1}{2} + \varepsilon})$ for all $\varepsilon > 0$ as the real number $x$ defining the sequence tends to infinity. The Farey sequences themselves are rational numbers less than 1, and fit onto the binary Farey tree described by Vepstas. The ends of the infinite tree fit onto the boundary of the Poincare disc, when the modular domain view is mapped there. Thus the interval markers above may be exchanged for roots of unity on the unit circle, and these compared to the leaves of the Farey tree.

Kauffman et al (p 51) show that this version of the Riemann hypothesis is equivalent to a question about messy unknots. They also look at DNA recombination. Unknots described by rational tangles are labelled by the pairs of adjacent rationals in a Farey sequence. So two tangles labelling two adjacent leaves of the tree at infinity can be used to construct unknots.

[1] H. M. Edwards, Riemann’s Zeta Function, Academic Press (1974)


2 Responses so far »

  1. 1

    Matti Pitkanen said,

    Dear Kea,

    you say that Riemann hypothesis is equivalent to maximally even distribution of Farey numbers in unit interval. This seems to be equivalent with a similar property for the distribution of roots of unity at unit circle (you mentioned map of unit interval to unit half circle). If so, these evenly distributed phases would provide the best possible discrete approximation to the full continuum of phases as algebraic numbers.

    In TGD framework the roots of unity define quantum phases q in the hierarchy of Jones inclusions and sectors of generalized imbedding space. Roots of unity define also the the phases representable in the framework of algebraic physics unifying real and p-adic physics to singlet coherent whole.

    The Platonia where RH holds true would be the best possible world in the sense that algebraic physics which is behind cognitive representations would allow the best possible approximation to the continuum physics (both for numbers in unit interval and for phases on unit circle). Platonia with RH would be cognitive paradise;-).

  2. 2

    Kea said,

    Platonia with RH would be cognitive paradise.

    Well, as Pangloss would have said, it must be true then!

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