Skip to content →

Tag: Riemann

New world record obscurification

I’ve always thought of Alain Connes as the unchallengeable world-champion opaque mathematical writing, but then again, I was proven wrong.

Alain’s writings are crystal clear compared to the monstrosity the AMS released to the world : In search of the Riemann zeros – Strings, fractal membranes and noncommutative spacetimes by Michel L. Lapidus.

Here’s a generic half-page from a total of 558 pages (or rather 314, as the remainder consists of appendices, bibliography and indices…). I couldn’t find a single precise, well-defined and proven statement in the entire book.

4.2. Fractal Membranes and the Second Quantization of Fractal Strings
“The first quantization is a mystery while the second quantization is a functor” Edward Nelson (quoted in [Con6,p.515])

We briefly discuss here joint work in preparation with Ryszard Nest [LapNe1]. This work was referred to several times in Chapter 3, and, as we pointed out there, it provides mathematically rigorous construction of fractal membranes (as well as of self-similar membranes), in the spirit of noncommutative geometry and quantum field theory (as well as of string theory). It also enables us to show that the expected properties of fractal (or self-similar) membranes, derived in our semi-heuristic model presented in Sections 3.2 and 3.2. are actually satisfied by the rigorous model in [LapNe1]. In particular, there is a surprisingly good agreement between the author’s original intuition on fractal (or self-similar) membrane, conceived as an (adelic) Riemann surface with infinite genus or as an (adelic) infinite dimensional torus, and properties of the noncommutative geometric model in [LapNe1]. In future joint work, we hope to go beyond [LapNe1] and to give even more (noncommutative) geometric content to this analogy, possibly along the lines suggested in the next section (4.3).
We will merely outline some aspects of the construction, without supplying any technical details, instead referring the interested reader to the forthcoming paper [LapNe1] for a complete exposition of the construction and precise statements of results.

Can the AMS please explain to the interested person buying this book why (s)he will have to await a (possible) forthcoming paper to (hopefully) make some sense of this apparent nonsense?

3 Comments

KMS, Gibbs & zeta function

Time to wrap up this series on the Bost-Connes algebra. Here’s what we have learned so far : the convolution product on double cosets

$\begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} \backslash \begin{bmatrix} 1 & \mathbb{Q} \\ 0 & \mathbb{Q}_{> 0} \end{bmatrix} / \begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} $

is a noncommutative algebra, the Bost-Connes Hecke algebra $\mathcal{H} $, which is a bi-chrystalline graded algebra (somewhat weaker than ‘strongly graded’) with part of degree one the group-algebra $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] $. Further, $\mathcal{H} $ has a natural one-parameter family of algebra automorphisms $\sigma_t $ defined by $\sigma_t(X_n) = n^{it}X_n $ and $\sigma_t(Y_{\lambda})=Y_{\lambda} $.

For any algebra $A $ together with a one-parameter family of automorphisms $\sigma_t $ one is interested in KMS-states or Kubo-Martin-Schwinger states with parameter $\beta $, $KMS_{\beta} $ (this parameter is often called the ‘invers temperature’ of the system) as these are suitable equilibria states. Recall that a state is a special linear functional $\phi $ on $A $ (in particular it must have norm one) and it belongs to $KMS_{\beta} $ if the following commutation relation holds for all elements $a,b \in A $

$\phi(a \sigma_{i\beta}(b)) = \phi(b a) $

Let us work out the special case when $A $ is the matrix-algebra $M_n(\mathbb{C}) $. To begin, all algebra-automorphisms are inner in this case, so any one-parameter family of automorphisms is of the form

$\sigma_t(a) = e^{itH} a e^{-itH} $

where $e^{itH} $ is the matrix-exponential of the nxn matrix $H $. For any parameter $\beta $ we claim that the linear functional

$\phi(a) = \frac{1}{tr(e^{-\beta H})} tr(a e^{-\beta H}) $

is a KMS-state.Indeed, we have for all matrices $a,b \in M_n(\mathbb{C}) $ that

$\phi(a \sigma_{i \beta}(b)) = \frac{1}{tr(e^{-\beta H})} tr(a e^{- \beta H} b e^{\beta H} e^{- \beta H}) $

$= \frac{1}{tr(e^{-\beta H})} tr(a e^{-\beta H} b) = \frac{1}{tr(e^{-\beta H})} tr(ba e^{-\beta H}) = \phi(ba) $

(the next to last equality follows from cyclic-invariance of the trace map).
These states are usually called Gibbs states and the normalization factor $\frac{1}{tr(e^{-\beta H})} $ (needed because a state must have norm one) is called the partition function of the system. Gibbs states have arisen from the study of ideal gases and the place to read up on all of this are the first two chapters of the second volume of “Operator algebras and quantum statistical mechanics” by Ola Bratelli and Derek Robinson.

This gives us a method to construct KMS-states for an arbitrary algebra $A $ with one-parameter automorphisms $\sigma_t $ : take a simple n-dimensional representation $\pi~:~A \mapsto M_n(\mathbb{C}) $, find the matrix $H $ determining the image of the automorphisms $\pi(\sigma_t) $ and take the Gibbs states as defined before.

Let us return now to the Bost-Connes algebra $\mathcal{H} $. We don’t know any finite dimensional simple representations of $\mathcal{H} $ but, sure enough, have plenty of graded simple representations. By the usual strongly-graded-yoga they should correspond to simple finite dimensional representations of the part of degree one $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] $ (all of them being one-dimensional and corresponding to characters of $\mathbb{Q}/\mathbb{Z} $).

Hence, for any $u \in \mathcal{G} = \prod_p \hat{\mathbb{Z}}_p^{\ast} $ (details) we have a graded simple $\mathcal{H} $-representation $S_u = \oplus_{n \in \mathbb{N}_+} \mathbb{C} e_n $ with action defined by

$\begin{cases} \pi_u(X_n)(e_m) = e_{nm} \\ \pi_u(Y_{\lambda})(e_m) = e^{2\pi i n u . \lambda} e_m \end{cases} $

Here, $u.\lambda $ is computed using the ‘chinese-remainder-identification’ $\mathcal{A}/\mathcal{R} = \mathbb{Q}/\mathbb{Z} $ (details).

Even when the representations $S_u $ are not finite dimensional, we can mimic the above strategy : we should find a linear operator $H $ determining the images of the automorphisms $\pi_u(\sigma_t) $. We claim that the operator is defined by $H(e_n) = log(n) e_n $ for all $n \in \mathbb{N}_+ $. That is, we claim that for elements $a \in \mathcal{H} $ we have

$\pi_u(\sigma_t(a)) = e^{itH} \pi_u(a) e^{-itH} $

So let us compute the action of both sides on $e_m $ when $a=X_n $. The left hand side gives $\pi_u(n^{it}X_n)(e_m) = n^{it} e_{mn} $ whereas the right-hand side becomes

$e^{itH}\pi_u(X_n) e^{-itH}(e_m) = e^{itH} \pi_u(X_n) m^{-it} e_m = $

$e^{itH} m^{-it} e_{mn} = (mn)^{it} m^{-it} e_{mn} = n^{it} e_{mn} $

proving the claim. For any parameter $\beta $ this then gives us a KMS-state for the Bost-Connes algebra by

$\phi_u(a) = \frac{1}{Tr(e^{-\beta H})} Tr(\pi_u(a) e^{-\beta H}) $

Finally, let us calculate the normalization factor (or partition function) $\frac{1}{Tr(e^{-\beta H})} $. Because $e^{-\beta H}(e_n) = n^{-\beta} e_n $ we have for that the trace

$Tr(e^{-\beta H}) = \sum_{n \in \mathbb{N}_+} \frac{1}{n^{\beta}} = \zeta(\beta) $

is equal to the Riemann zeta-value $\zeta(\beta) $ (at least when $\beta > 1 $).

Summarizing, we started with the definition of the Bost-Connes algebra $\mathcal{H} $, found a canonical one-parameter subgroup of algebra automorphisms $\sigma_t $ and computed that the natural equilibria states with respect to this ‘time evolution’ have as their partition function the Riemann zeta-function. Voila!

2 Comments

music of the primes (1)

This semester, I’m running a 3rd year course on Marcus du Sautoy’s The music of the primes. The concept being that students may suggest topics, merely sketched in the book, and then we’ll go a little deeper into them.

I’ve been rather critical about the book before, but, rereading it last week (and knowing a bit better the limitations of bringing mathematics to the masses…) I think du Sautoy did a great job. Sure, it focusses too much on people and places and too little on mathematics, but that goes with the format.

I wanted to start off gently by playing the open-university dvd-series so that students would have a very rough outline of the book from the very start (as well as a mental image to some of the places mentioned, such as Bletchley Park, the IAS, Gottingen…). However, the vagueness of it all seemed to work on their nerves … in particular the trumpet scenes

Afterwards, they demanded that I should explain next week what on earth the zeroes of the Riemann zeta function had to do with counting primes and what all this nonsensical ‘music of the primes’ was about.

Well, here is the genuine music of the primes (taken from the Riemann page by Jeffrey Stopple whose excellent introductory text A Primer of Analytic Number Theory I’ll use to show them some concrete stuff (they have their first course on complex analysis also this semester, so I cannot go too deep into it).




Jeffrey writes “This sound is best listened to with headphones or external speakers. For maximum effect, play it LOUD.” But, what is the story behind it?

The Von Mangoldt function $\Lambda(n) $ assigns $log(p) $ whenever $n=p^k $ is a prime power and zero otherwise. One can then consider the function

$\Psi(x) = \frac{1}{2}(\sum_{n < x} \Lambda(n) + \sum_{n \leq x} \Lambda(n)) $

which makes a jump at prime power values and the jump-size depends on the prime. Here is a graph of its small values

It’s not quite the function $\pi(x) $ (counting the number of primes smaller than x) but it sure contains enough information to obtain this provided we have a way of describing $\Psi(x) $.

The Riemann zeta function (or rather $~(s-1)\zeta(s) $) has two product descriptions, the Hadamard product formula (running over all zeroes, both the trivial ones at $-2n $ and those in the critical strip), which is valid for all complex s and the Euler product valid for all $Re(s) > 1 $. This will allow us to calculate in two different ways $\zeta'(s)/\zeta(s) $ which in turn allows us to have an explicit description of $\Psi(s) $ known as the Von Mangoldt formula

$\Psi(x) = x – \frac{1}{2}log(1 – \frac{1}{x^2}) – log(2 \pi) – \sum_{\rho} \frac{x^{\rho}}{\rho} $

where only the last term depends on the zeta-zeroes $\rho $ lying in the critical strip (and conjecturally all lying on the line $Re(x) = \frac{1}{2} $. The first few terms (those independent of the zeroes) give a continuous approximation of $\Psi(x) $ but how on earth can we get from that approxamation (on the left) to the step-like function itself (on the right)?



We can group together zeta-zeroes $\rho=\beta + i \gamma $ with their comlex conjugate zeroes $\overline{\rho} $ and then one shows that the attribution to the Von Mangoldt formula is

$\frac{x^{\rho}}{\rho} + \frac{x^{\overline{\rho}}}{\overline{rho}} = \frac{2 x^{\beta}}{| \rho | }cos(\gamma log(x) – arctan(\gamma/\beta)) $

Ignoring the term $x^{\beta} $ this is a peridodic function with amplitude $2/| \rho | $ (so getting smaller for larger and larger zeroes) and period $2\pi/ \gamma $. If the Riemann hypothesis holds (meaning that $\beta=1/2 $ for all zeroes) one can even split a term in this contribution of every zero as a sort of ‘universal amplitude’. What is left is then a sum of purely periodic functions which a physicist will view as a superposition of (sound) waves and that is the music played by the primes!

Below, a video of the influence of adding the first 100 zeroes to a better and better approximation of $\Psi(x) $ (again taken from the Riemann page by Jeffrey Stopple). Surely watching the video will convince anyone of the importance of the Riemann zeta-zeroes to the prime-counting problem..




One Comment